Espacio vectorial - Wikipedia, la enciclopedia libre

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cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Definición</span> </button> <ul id="toc-Definición-sublist" class="vector-toc-list"> <li id="toc-Observaciones" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Observaciones"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Observaciones</span> </div> </a> <ul id="toc-Observaciones-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propiedades" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Propiedades"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Propiedades</span> </div> </a> <ul id="toc-Propiedades-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primer_ejemplo_con_demostración" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primer_ejemplo_con_demostración"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Primer ejemplo con demostración</span> </div> </a> <ul id="toc-Primer_ejemplo_con_demostración-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ejemplos" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Ejemplos"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Ejemplos</span> </div> </a> <button aria-controls="toc-Ejemplos-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Ejemplos</span> </button> <ul id="toc-Ejemplos-sublist" class="vector-toc-list"> <li id="toc-Los_cuerpos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Los_cuerpos"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Los cuerpos</span> </div> </a> <ul id="toc-Los_cuerpos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sucesiones_sobre_un_cuerpo_K" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sucesiones_sobre_un_cuerpo_K"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Sucesiones sobre un cuerpo K</span> </div> </a> <ul id="toc-Sucesiones_sobre_un_cuerpo_K-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Espacios_de_aplicaciones_sobre_un_cuerpo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Espacios_de_aplicaciones_sobre_un_cuerpo"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Espacios de aplicaciones sobre un cuerpo</span> </div> </a> <ul id="toc-Espacios_de_aplicaciones_sobre_un_cuerpo-sublist" class="vector-toc-list"> <li id="toc-Los_polinomios" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Los_polinomios"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.1</span> <span>Los polinomios</span> </div> </a> <ul id="toc-Los_polinomios-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Funciones_trigonométricas" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Funciones_trigonométricas"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.2</span> <span>Funciones trigonométricas</span> </div> </a> <ul id="toc-Funciones_trigonométricas-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Los_sistemas_de_ecuaciones_lineales_homogéneas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Los_sistemas_de_ecuaciones_lineales_homogéneas"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Los sistemas de ecuaciones lineales homogéneas</span> </div> </a> <ul id="toc-Los_sistemas_de_ecuaciones_lineales_homogéneas-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Subespacio_vectorial" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Subespacio_vectorial"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Subespacio vectorial</span> </div> </a> <button aria-controls="toc-Subespacio_vectorial-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Subespacio vectorial</span> </button> <ul id="toc-Subespacio_vectorial-sublist" class="vector-toc-list"> <li id="toc-Definición_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definición_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Definición</span> </div> </a> <ul id="toc-Definición_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Consecuencias" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Consecuencias"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Consecuencias</span> </div> </a> <ul id="toc-Consecuencias-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Resultados_internos" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Resultados_internos"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Resultados internos</span> </div> </a> <button aria-controls="toc-Resultados_internos-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Resultados internos</span> </button> <ul id="toc-Resultados_internos-sublist" class="vector-toc-list"> <li id="toc-Combinación_lineal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Combinación_lineal"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Combinación lineal</span> </div> </a> <ul id="toc-Combinación_lineal-sublist" class="vector-toc-list"> <li id="toc-Proposición_1" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Proposición_1"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.1</span> <span>Proposición 1</span> </div> </a> <ul id="toc-Proposición_1-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Independencia_lineal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Independencia_lineal"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Independencia lineal</span> </div> </a> <ul id="toc-Independencia_lineal-sublist" class="vector-toc-list"> <li id="toc-Proposición_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Proposición_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.1</span> <span>Proposición 2</span> </div> </a> <ul id="toc-Proposición_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Base_de_un_espacio_vectorial" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Base_de_un_espacio_vectorial"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Base de un espacio vectorial</span> </div> </a> <ul id="toc-Base_de_un_espacio_vectorial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Base_formalmente" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Base_formalmente"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Base formalmente</span> </div> </a> <ul id="toc-Base_formalmente-sublist" class="vector-toc-list"> <li id="toc-Teorema_de_la_base_de_generadores" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Teorema_de_la_base_de_generadores"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4.1</span> <span>Teorema de la base de generadores</span> </div> </a> <ul id="toc-Teorema_de_la_base_de_generadores-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teorema_Steinitz" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Teorema_Steinitz"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4.2</span> <span>Teorema Steinitz</span> </div> </a> <ul id="toc-Teorema_Steinitz-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Observación" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Observación"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Observación</span> </div> </a> <ul id="toc-Observación-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dimensión" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dimensión"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.6</span> <span>Dimensión</span> </div> </a> <ul id="toc-Dimensión-sublist" class="vector-toc-list"> <li id="toc-Notación_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Notación_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.6.1</span> <span>Notación</span> </div> </a> <ul id="toc-Notación_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Intersección_de_subespacios_vectoriales" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Intersección_de_subespacios_vectoriales"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.7</span> <span>Intersección de subespacios vectoriales</span> </div> </a> <ul id="toc-Intersección_de_subespacios_vectoriales-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Suma_de_subespacios_vectoriales" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Suma_de_subespacios_vectoriales"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.8</span> <span>Suma de subespacios vectoriales</span> </div> </a> <ul id="toc-Suma_de_subespacios_vectoriales-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teorema_Fórmula_de_Grassmann" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Teorema_Fórmula_de_Grassmann"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.9</span> <span>Teorema Fórmula de Grassmann</span> </div> </a> <ul id="toc-Teorema_Fórmula_de_Grassmann-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Suma_directa_de_subespacios_vectoriales" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Suma_directa_de_subespacios_vectoriales"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.10</span> <span>Suma directa de subespacios vectoriales</span> </div> </a> <ul id="toc-Suma_directa_de_subespacios_vectoriales-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cociente_de_espacios_vectoriales" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cociente_de_espacios_vectoriales"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.11</span> <span>Cociente de espacios vectoriales</span> </div> </a> <ul id="toc-Cociente_de_espacios_vectoriales-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Construcciones_básicas" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Construcciones_básicas"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Construcciones básicas</span> </div> </a> <button aria-controls="toc-Construcciones_básicas-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Construcciones básicas</span> </button> <ul id="toc-Construcciones_básicas-sublist" class="vector-toc-list"> <li id="toc-Suma_directa_de_espacios_vectoriales" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Suma_directa_de_espacios_vectoriales"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Suma directa de espacios vectoriales</span> </div> </a> <ul id="toc-Suma_directa_de_espacios_vectoriales-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Espacios_vectoriales_con_estructura_adicional" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Espacios_vectoriales_con_estructura_adicional"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Espacios vectoriales con estructura adicional</span> </div> </a> <button aria-controls="toc-Espacios_vectoriales_con_estructura_adicional-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Espacios vectoriales con estructura adicional</span> </button> <ul id="toc-Espacios_vectoriales_con_estructura_adicional-sublist" class="vector-toc-list"> <li id="toc-Espacios_normados" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Espacios_normados"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Espacios normados</span> </div> </a> <ul id="toc-Espacios_normados-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Espacios_vectoriales_topológicos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Espacios_vectoriales_topológicos"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Espacios vectoriales topológicos</span> </div> </a> <ul id="toc-Espacios_vectoriales_topológicos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Espacios_de_Banach" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Espacios_de_Banach"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Espacios de Banach</span> </div> </a> <ul id="toc-Espacios_de_Banach-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Espacios_prehilbertianos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Espacios_prehilbertianos"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.4</span> <span>Espacios prehilbertianos</span> </div> </a> <ul id="toc-Espacios_prehilbertianos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Espacios_de_Hilbert" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Espacios_de_Hilbert"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.5</span> <span>Espacios de Hilbert</span> </div> </a> <ul id="toc-Espacios_de_Hilbert-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Morfismos_entre_espacios_vectoriales" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Morfismos_entre_espacios_vectoriales"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Morfismos entre espacios vectoriales</span> </div> </a> <button aria-controls="toc-Morfismos_entre_espacios_vectoriales-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Morfismos entre espacios vectoriales</span> </button> <ul id="toc-Morfismos_entre_espacios_vectoriales-sublist" class="vector-toc-list"> <li id="toc-Aplicaciones_lineales" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aplicaciones_lineales"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Aplicaciones lineales</span> </div> </a> <ul id="toc-Aplicaciones_lineales-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Véase_también" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Véase_también"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Véase también</span> </div> </a> <ul id="toc-Véase_también-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referencias" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Referencias"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Referencias</span> </div> </a> <button aria-controls="toc-Referencias-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Referencias</span> </button> <ul id="toc-Referencias-sublist" class="vector-toc-list"> <li id="toc-Notas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notas"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>Notas</span> </div> </a> <ul id="toc-Notas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referencias_históricas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Referencias_históricas"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.2</span> <span>Referencias históricas</span> </div> </a> <ul id="toc-Referencias_históricas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografía" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliografía"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.3</span> <span>Bibliografía</span> </div> </a> <ul id="toc-Bibliografía-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Enlaces_externos" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Enlaces_externos"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Enlaces externos</span> </div> </a> <ul id="toc-Enlaces_externos-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contenidos" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Cambiar a la tabla de contenidos" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Cambiar a la tabla de contenidos</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Espacio vectorial</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ir a un artículo en otro idioma. Disponible en 77 idiomas" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-77" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">77 idiomas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Vektorruimte" title="Vektorruimte (afrikáans)" lang="af" hreflang="af" data-title="Vektorruimte" data-language-autonym="Afrikaans" data-language-local-name="afrikáans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%D8%A1_%D9%85%D8%AA%D8%AC%D9%87%D9%8A" title="فضاء متجهي (árabe)" lang="ar" hreflang="ar" data-title="فضاء متجهي" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Espaciu_vectorial" title="Espaciu vectorial (asturiano)" lang="ast" hreflang="ast" data-title="Espaciu vectorial" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BB%D1%8B_%D0%B0%D1%80%D0%B0%D1%83%D1%8B%D2%A1" title="Векторлы арауыҡ (baskir)" lang="ba" hreflang="ba" data-title="Векторлы арауыҡ" data-language-autonym="Башҡортса" data-language-local-name="baskir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%B0%D1%80%D0%BD%D0%B0%D1%8F_%D0%BF%D1%80%D0%B0%D1%81%D1%82%D0%BE%D1%80%D0%B0" title="Вектарная прастора (bielorruso)" lang="be" hreflang="be" data-title="Вектарная прастора" data-language-autonym="Беларуская" data-language-local-name="bielorruso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Линейно пространство (búlgaro)" lang="bg" hreflang="bg" data-title="Линейно пространство" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%B5%E0%A5%87%E0%A4%95%E0%A5%8D%E0%A4%9F%E0%A4%B0_%E0%A4%B8%E0%A5%8D%E0%A4%AA%E0%A5%87%E0%A4%B8" title="वेक्टर स्पेस (Bhojpuri)" lang="bh" hreflang="bh" data-title="वेक्टर स्पेस" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%A6%E0%A6%BF%E0%A6%95_%E0%A6%B0%E0%A6%BE%E0%A6%B6%E0%A6%BF%E0%A6%B0_%E0%A6%AC%E0%A7%80%E0%A6%9C%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="সদিক রাশির বীজগণিত (bengalí)" lang="bn" hreflang="bn" data-title="সদিক রাশির বীজগণিত" data-language-autonym="বাংলা" data-language-local-name="bengalí" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Vektorski_prostor" title="Vektorski prostor (bosnio)" lang="bs" hreflang="bs" data-title="Vektorski prostor" data-language-autonym="Bosanski" data-language-local-name="bosnio" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca badge-Q17437796 badge-featuredarticle mw-list-item" title="artículo destacado"><a href="https://ca.wikipedia.org/wiki/Espai_vectorial" title="Espai vectorial (catalán)" lang="ca" hreflang="ca" data-title="Espai vectorial" data-language-autonym="Català" data-language-local-name="catalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%DB%86%D8%B4%D8%A7%DB%8C%DB%8C%DB%8C_%D8%A6%D8%A7%DA%95%D8%A7%D8%B3%D8%AA%DB%95%D8%A8%DA%95%DB%95%DA%A9%D8%A7%D9%86" title="بۆشاییی ئاڕاستەبڕەکان (kurdo sorani)" lang="ckb" hreflang="ckb" data-title="بۆشاییی ئاڕاستەبڕەکان" data-language-autonym="کوردی" data-language-local-name="kurdo sorani" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vektorov%C3%BD_prostor" title="Vektorový prostor (checo)" lang="cs" hreflang="cs" data-title="Vektorový prostor" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BB%D0%B0_%D1%83%C3%A7%D0%BB%C4%83%D1%85" title="Векторла уçлăх (chuvasio)" lang="cv" hreflang="cv" data-title="Векторла уçлăх" data-language-autonym="Чӑвашла" data-language-local-name="chuvasio" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gofod_fector" title="Gofod fector (galés)" lang="cy" hreflang="cy" data-title="Gofod fector" data-language-autonym="Cymraeg" data-language-local-name="galés" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Vektorrum" title="Vektorrum (danés)" lang="da" hreflang="da" data-title="Vektorrum" data-language-autonym="Dansk" data-language-local-name="danés" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Vektorraum" title="Vektorraum (alemán)" lang="de" hreflang="de" data-title="Vektorraum" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B9%CE%B1%CE%BD%CF%85%CF%83%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CF%8C%CF%82_%CF%87%CF%8E%CF%81%CE%BF%CF%82" title="Διανυσματικός χώρος (griego)" lang="el" hreflang="el" data-title="Διανυσματικός χώρος" data-language-autonym="Ελληνικά" data-language-local-name="griego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437798 badge-goodarticle mw-list-item" title="artículo bueno"><a href="https://en.wikipedia.org/wiki/Vector_space" title="Vector space (inglés)" lang="en" hreflang="en" data-title="Vector space" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Vektora_spaco" title="Vektora spaco (esperanto)" lang="eo" hreflang="eo" data-title="Vektora spaco" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Vektorruum" title="Vektorruum (estonio)" lang="et" hreflang="et" data-title="Vektorruum" data-language-autonym="Eesti" data-language-local-name="estonio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bektore_espazio" title="Bektore espazio (euskera)" lang="eu" hreflang="eu" data-title="Bektore espazio" data-language-autonym="Euskara" data-language-local-name="euskera" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%DB%8C_%D8%A8%D8%B1%D8%AF%D8%A7%D8%B1%DB%8C" title="فضای برداری (persa)" lang="fa" hreflang="fa" data-title="فضای برداری" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Vektoriavaruus" title="Vektoriavaruus (finés)" lang="fi" hreflang="fi" data-title="Vektoriavaruus" data-language-autonym="Suomi" data-language-local-name="finés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Espace_vectoriel" title="Espace vectoriel (francés)" lang="fr" hreflang="fr" data-title="Espace vectoriel" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Sp%C3%A1s_veicteoireach" title="Spás veicteoireach (irlandés)" lang="ga" hreflang="ga" data-title="Spás veicteoireach" data-language-autonym="Gaeilge" data-language-local-name="irlandés" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Espazo_vectorial" title="Espazo vectorial (gallego)" lang="gl" hreflang="gl" data-title="Espazo vectorial" data-language-autonym="Galego" data-language-local-name="gallego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99" title="מרחב וקטורי (hebreo)" lang="he" hreflang="he" data-title="מרחב וקטורי" data-language-autonym="עברית" data-language-local-name="hebreo" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%A6%E0%A4%BF%E0%A4%B6_%E0%A4%AC%E0%A5%80%E0%A4%9C%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="सदिश बीजगणित (hindi)" lang="hi" hreflang="hi" data-title="सदिश बीजगणित" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Vektorski_prostor" title="Vektorski prostor (croata)" lang="hr" hreflang="hr" data-title="Vektorski prostor" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Vektort%C3%A9r" title="Vektortér (húngaro)" lang="hu" hreflang="hu" data-title="Vektortér" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8E%D5%A5%D5%AF%D5%BF%D5%B8%D6%80%D5%A1%D5%AF%D5%A1%D5%B6_%D5%BF%D5%A1%D6%80%D5%A1%D5%AE%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Վեկտորական տարածություն (armenio)" lang="hy" hreflang="hy" data-title="Վեկտորական տարածություն" data-language-autonym="Հայերեն" data-language-local-name="armenio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Spatio_vectorial" title="Spatio vectorial (interlingua)" lang="ia" hreflang="ia" data-title="Spatio vectorial" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ruang_vektor" title="Ruang vektor (indonesio)" lang="id" hreflang="id" data-title="Ruang vektor" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Vigurr%C3%BAm" title="Vigurrúm (islandés)" lang="is" hreflang="is" data-title="Vigurrúm" data-language-autonym="Íslenska" data-language-local-name="islandés" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_vettoriale" title="Spazio vettoriale (italiano)" lang="it" hreflang="it" data-title="Spazio vettoriale" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E7%A9%BA%E9%96%93" title="ベクトル空間 (japonés)" lang="ja" hreflang="ja" data-title="ベクトル空間" data-language-autonym="日本語" data-language-local-name="japonés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B2%A1%ED%84%B0_%EA%B3%B5%EA%B0%84" title="벡터 공간 (coreano)" lang="ko" hreflang="ko" data-title="벡터 공간" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%B4%D1%83%D0%BA_%D0%BC%D0%B5%D0%B9%D0%BA%D0%B8%D0%BD%D0%B4%D0%B8%D0%BA" title="Вектордук мейкиндик (kirguís)" lang="ky" hreflang="ky" data-title="Вектордук мейкиндик" data-language-autonym="Кыргызча" data-language-local-name="kirguís" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Spatium_vectoriale" title="Spatium vectoriale (latín)" lang="la" hreflang="la" data-title="Spatium vectoriale" data-language-autonym="Latina" data-language-local-name="latín" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Spazzi_vettorial" title="Spazzi vettorial (lombardo)" lang="lmo" hreflang="lmo" data-title="Spazzi vettorial" data-language-autonym="Lombard" data-language-local-name="lombardo" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BB%80%E0%BA%A7%E0%BA%B1%E0%BA%81%E0%BB%80%E0%BA%95%E0%BA%B5" title="ເວັກເຕີ (lao)" lang="lo" hreflang="lo" data-title="ເວັກເຕີ" data-language-autonym="ລາວ" data-language-local-name="lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Vektorin%C4%97_erdv%C4%97" title="Vektorinė erdvė (lituano)" lang="lt" hreflang="lt" data-title="Vektorinė erdvė" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Vektoru_telpa" title="Vektoru telpa (letón)" lang="lv" hreflang="lv" data-title="Vektoru telpa" data-language-autonym="Latviešu" data-language-local-name="letón" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Векторски простор (macedonio)" lang="mk" hreflang="mk" data-title="Векторски простор" data-language-autonym="Македонски" data-language-local-name="macedonio" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B4%A6%E0%B4%BF%E0%B4%B6%E0%B4%B8%E0%B4%AE%E0%B4%B7%E0%B5%8D%E0%B4%9F%E0%B4%BF" title="സദിശസമഷ്ടി (malayálam)" lang="ml" hreflang="ml" data-title="സദിശസമഷ്ടി" data-language-autonym="മലയാളം" data-language-local-name="malayálam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Ruang_vektor" title="Ruang vektor (malayo)" lang="ms" hreflang="ms" data-title="Ruang vektor" data-language-autonym="Bahasa Melayu" data-language-local-name="malayo" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vectorruimte" title="Vectorruimte (neerlandés)" lang="nl" hreflang="nl" data-title="Vectorruimte" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Vektorrom" title="Vektorrom (noruego nynorsk)" lang="nn" hreflang="nn" data-title="Vektorrom" data-language-autonym="Norsk nynorsk" data-language-local-name="noruego nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Vektorrom" title="Vektorrom (noruego bokmal)" lang="nb" hreflang="nb" data-title="Vektorrom" data-language-autonym="Norsk bokmål" data-language-local-name="noruego bokmal" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Espaci_vectoriau" title="Espaci vectoriau (occitano)" lang="oc" hreflang="oc" data-title="Espaci vectoriau" data-language-autonym="Occitan" data-language-local-name="occitano" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B5%E0%A9%88%E0%A8%95%E0%A8%9F%E0%A8%B0_%E0%A8%B8%E0%A8%AA%E0%A9%87%E0%A8%B8" title="ਵੈਕਟਰ ਸਪੇਸ (punyabí)" lang="pa" hreflang="pa" data-title="ਵੈਕਟਰ ਸਪੇਸ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punyabí" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przestrze%C5%84_liniowa" title="Przestrzeń liniowa (polaco)" lang="pl" hreflang="pl" data-title="Przestrzeń liniowa" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Spassi_vetorial" title="Spassi vetorial (Piedmontese)" lang="pms" hreflang="pms" data-title="Spassi vetorial" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%88%DB%8C%DA%A9%D9%B9%D8%B1_%D8%B3%D9%BE%DB%8C%D8%B3" title="ویکٹر سپیس (Western Punjabi)" lang="pnb" hreflang="pnb" data-title="ویکٹر سپیس" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_vetorial" title="Espaço vetorial (portugués)" lang="pt" hreflang="pt" data-title="Espaço vetorial" data-language-autonym="Português" data-language-local-name="portugués" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro badge-Q17437798 badge-goodarticle mw-list-item" title="artículo bueno"><a href="https://ro.wikipedia.org/wiki/Spa%C8%9Biu_vectorial" title="Spațiu vectorial (rumano)" lang="ro" hreflang="ro" data-title="Spațiu vectorial" data-language-autonym="Română" data-language-local-name="rumano" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Векторное пространство (ruso)" lang="ru" hreflang="ru" data-title="Векторное пространство" data-language-autonym="Русский" data-language-local-name="ruso" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Spazziu_vitturiali" title="Spazziu vitturiali (siciliano)" lang="scn" hreflang="scn" data-title="Spazziu vitturiali" data-language-autonym="Sicilianu" data-language-local-name="siciliano" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Vektorski_prostor" title="Vektorski prostor (serbocroata)" lang="sh" hreflang="sh" data-title="Vektorski prostor" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Vector_space" title="Vector space (Simple English)" lang="en-simple" hreflang="en-simple" data-title="Vector space" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Vektorov%C3%BD_priestor" title="Vektorový priestor (eslovaco)" lang="sk" hreflang="sk" data-title="Vektorový priestor" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Vektorski_prostor" title="Vektorski prostor (esloveno)" lang="sl" hreflang="sl" data-title="Vektorski prostor" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Hap%C3%ABsira_vektoriale" title="Hapësira vektoriale (albanés)" lang="sq" hreflang="sq" data-title="Hapësira vektoriale" data-language-autonym="Shqip" data-language-local-name="albanés" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Векторски простор (serbio)" lang="sr" hreflang="sr" data-title="Векторски простор" data-language-autonym="Српски / srpski" data-language-local-name="serbio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Linj%C3%A4rt_rum" title="Linjärt rum (sueco)" lang="sv" hreflang="sv" data-title="Linjärt rum" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AE%BF%E0%AE%9A%E0%AF%88%E0%AE%AF%E0%AE%A9%E0%AF%8D_%E0%AE%B5%E0%AF%86%E0%AE%B3%E0%AE%BF" title="திசையன் வெளி (tamil)" lang="ta" hreflang="ta" data-title="திசையன் வெளி" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Espasyong_bektor" title="Espasyong bektor (tagalo)" lang="tl" hreflang="tl" data-title="Espasyong bektor" data-language-autonym="Tagalog" data-language-local-name="tagalo" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Vekt%C3%B6r_uzay%C4%B1" title="Vektör uzayı (turco)" lang="tr" hreflang="tr" data-title="Vektör uzayı" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BD%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Векторний простір (ucraniano)" lang="uk" hreflang="uk" data-title="Векторний простір" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B3%D9%85%D8%AA%DB%8C%DB%81_%D9%85%DA%A9%D8%A7%DA%BA" title="سمتیہ مکاں (urdu)" lang="ur" hreflang="ur" data-title="سمتیہ مکاں" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Spasio_vetorial" title="Spasio vetorial (Venetian)" lang="vec" hreflang="vec" data-title="Spasio vetorial" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kh%C3%B4ng_gian_vect%C6%A1" title="Không gian vectơ (vietnamita)" lang="vi" hreflang="vi" data-title="Không gian vectơ" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间 (chino wu)" lang="wuu" hreflang="wuu" data-title="向量空间" data-language-autonym="吴语" data-language-local-name="chino wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间 (chino)" lang="zh" hreflang="zh" data-title="向量空间" data-language-autonym="中文" data-language-local-name="chino" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%9F%A2%E9%87%8F%E7%A9%BA%E9%96%93" title="矢量空間 (Literary Chinese)" lang="lzh" hreflang="lzh" data-title="矢量空間" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Hi%C3%B2ng-li%C5%8Dng_khong-kan" title="Hiòng-liōng khong-kan (chino min nan)" lang="nan" hreflang="nan" data-title="Hiòng-liōng khong-kan" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="chino min nan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%96%93" title="向量空間 (cantonés)" lang="yue" hreflang="yue" data-title="向量空間" data-language-autonym="粵語" data-language-local-name="cantonés" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q125977#sitelinks-wikipedia" title="Editar enlaces interlingüísticos" class="wbc-editpage">Editar enlaces</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espacios de nombres"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Espacio_vectorial" title="Ver la página de contenido [c]" accesskey="c"><span>Artículo</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discusi%C3%B3n:Espacio_vectorial" rel="discussion" title="Discusión acerca de la página [t]" accesskey="t"><span>Discusión</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Cambiar variante de idioma" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">español</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Vistas"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Espacio_vectorial"><span>Leer</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Espacio_vectorial&action=edit" title="Editar esta página [e]" accesskey="e"><span>Editar</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Espacio_vectorial&action=history" title="Versiones anteriores de esta página [h]" accesskey="h"><span>Ver historial</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Página de herramientas"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Herramientas" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Herramientas</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Herramientas</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mover a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ocultar</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Más opciones" > <div class="vector-menu-heading"> Acciones </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Espacio_vectorial"><span>Leer</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Espacio_vectorial&action=edit" title="Editar esta página [e]" accesskey="e"><span>Editar</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Espacio_vectorial&action=history"><span>Ver historial</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:LoQueEnlazaAqu%C3%AD/Espacio_vectorial" title="Lista de todas las páginas de la wiki que enlazan aquí [j]" accesskey="j"><span>Lo que enlaza aquí</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:CambiosEnEnlazadas/Espacio_vectorial" rel="nofollow" title="Cambios recientes en las páginas que enlazan con esta [k]" accesskey="k"><span>Cambios en enlazadas</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=es" title="Subir archivos [u]" accesskey="u"><span>Subir archivo</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1ginasEspeciales" title="Lista de todas las páginas especiales [q]" accesskey="q"><span>Páginas especiales</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Espacio_vectorial&oldid=156217106" title="Enlace permanente a esta versión de la página"><span>Enlace permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Espacio_vectorial&action=info" title="Más información sobre esta página"><span>Información de la página</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citar&page=Espacio_vectorial&id=156217106&wpFormIdentifier=titleform" title="Información sobre cómo citar esta página"><span>Citar esta página</span></a></li><li id="t-urlshortener" 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</div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Página de herramientas"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apariencia"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Apariencia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover a la 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Para una introducción más accesible al concepto, véase <a href="/wiki/Vector" title="Vector">Vector</a></i></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Archivo:Vector_space_illust.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Vector_space_illust.svg/220px-Vector_space_illust.svg.png" decoding="async" width="220" height="269" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Vector_space_illust.svg/330px-Vector_space_illust.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Vector_space_illust.svg/440px-Vector_space_illust.svg.png 2x" data-file-width="454" data-file-height="555" /></a><figcaption>Representación artística de un espacio vectorial.</figcaption></figure> <p>En <a href="/wiki/%C3%81lgebra_lineal" title="Álgebra lineal">álgebra lineal</a>, un <b>espacio vectorial</b> (o también llamado <b>espacio lineal</b>) es una <a href="/wiki/Estructura_algebraica" title="Estructura algebraica">estructura algebraica</a> creada a partir de un <a href="/wiki/Dominio_de_definici%C3%B3n" class="mw-redirect" title="Dominio de definición">conjunto no vacío</a>, una <a href="/wiki/Operaci%C3%B3n_interna" title="Operación interna">operación interna</a> (llamada <i>suma</i>, definida para los elementos del conjunto) y una <a href="/wiki/Operaci%C3%B3n_externa" title="Operación externa">operación externa</a> (llamada <i>producto por un escalar</i>, definida entre dicho conjunto y otro conjunto, con estructura de <a href="/wiki/Cuerpo_(matem%C3%A1tica)" class="mw-redirect" title="Cuerpo (matemática)">cuerpo</a>) que satisface 8 propiedades fundamentales. </p><p>A los elementos de un espacio vectorial se les llama <a href="/wiki/Vector_(espacio_eucl%C3%ADdeo)" class="mw-redirect" title="Vector (espacio euclídeo)">vectores</a> y a los elementos del cuerpo se les conoce como <a href="/wiki/Escalar_(matem%C3%A1tica)" title="Escalar (matemática)">escalares</a> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historia">Historia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=1" title="Editar sección: Historia"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Históricamente, las primeras ideas que condujeron a los espacios vectoriales modernos se remontan al siglo <span style="font-variant:small-caps;text-transform:lowercase">XVII</span>: <a href="/wiki/Geometr%C3%ADa_anal%C3%ADtica" title="Geometría analítica">geometría analítica</a>, <a href="/wiki/Matriz_(matem%C3%A1tica)" title="Matriz (matemática)">matrices</a> y <a href="/wiki/Sistemas_de_ecuaciones_lineales" class="mw-redirect" title="Sistemas de ecuaciones lineales">sistemas de ecuaciones lineales</a>. </p><p>Los espacios vectoriales se derivan de la <a href="/wiki/Geometr%C3%ADa_af%C3%ADn" title="Geometría afín">geometría afín</a> a través de la introducción de <a href="/wiki/Coordenada" class="mw-redirect" title="Coordenada">coordenadas</a> en el plano o el espacio tridimensional. Alrededor de 1636, los matemáticos franceses <a href="/wiki/Descartes" class="mw-redirect" title="Descartes">Descartes</a> y <a href="/wiki/Fermat" class="mw-redirect" title="Fermat">Fermat</a> fundaron las bases de la <a href="/wiki/Geometr%C3%ADa_anal%C3%ADtica" title="Geometría analítica">geometría analítica</a> mediante la vinculación de las soluciones de una ecuación con dos variables a la determinación de una <a href="/wiki/Curva" title="Curva">curva</a> plana.<sup id="cite_ref-1" class="reference separada"><a href="#cite_note-1"><span class="corchete-llamada">[</span>nota 1<span class="corchete-llamada">]</span></a></sup> Para lograr una solución geométrica sin usar coordenadas, <a href="/wiki/Bernhard_Bolzano" class="mw-redirect" title="Bernhard Bolzano">Bernhard Bolzano</a> introdujo en 1804 ciertas operaciones sobre puntos, líneas y planos, que son predecesores de los vectores.<sup id="cite_ref-2" class="reference separada"><a href="#cite_note-2"><span class="corchete-llamada">[</span>nota 2<span class="corchete-llamada">]</span></a></sup> Este trabajo hizo uso del concepto de <a href="/wiki/Coordenadas_baric%C3%A9ntricas_(n-simplex)" title="Coordenadas baricéntricas (n-simplex)">coordenadas baricéntricas</a> de <a href="/wiki/August_Ferdinand_M%C3%B6bius" class="mw-redirect" title="August Ferdinand Möbius">August Ferdinand Möbius</a> de 1827.<sup id="cite_ref-3" class="reference separada"><a href="#cite_note-3"><span class="corchete-llamada">[</span>nota 3<span class="corchete-llamada">]</span></a></sup> </p><p>La primera formulación moderna y axiomática se debe a <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a>, a finales del siglo <span style="font-variant:small-caps;text-transform:lowercase">XIX</span>. Los siguientes avances en la teoría de espacios vectoriales provienen del <a href="/wiki/An%C3%A1lisis_funcional" title="Análisis funcional">análisis funcional</a>, principalmente de <a href="/wiki/Espacios_de_funciones" class="mw-redirect" title="Espacios de funciones">espacios de funciones</a>. Los problemas de Análisis funcional requerían resolver problemas sobre la <a href="/wiki/Convergencia_(matem%C3%A1ticas)" class="mw-redirect" title="Convergencia (matemáticas)">convergencia</a>. Esto se hizo dotando a los espacios vectoriales de una adecuada <a href="/wiki/Topolog%C3%ADa" title="Topología">topología</a>, permitiendo tener en cuenta cuestiones de <a href="/wiki/Proximidad" class="mw-redirect mw-disambig" title="Proximidad">proximidad</a> y <a href="/wiki/Funci%C3%B3n_continua" title="Función continua">continuidad</a>. Estos <a href="/wiki/Espacio_vectorial_topol%C3%B3gico" title="Espacio vectorial topológico">espacios vectoriales topológicos</a>, en particular los <a href="/wiki/Espacios_de_Banach" class="mw-redirect" title="Espacios de Banach">espacios de Banach</a> y los <a href="/wiki/Espacios_de_Hilbert" class="mw-redirect" title="Espacios de Hilbert">espacios de Hilbert</a> tienen una teoría más rica y elaborada. </p><p>El origen de la definición de los vectores es la definición de <a href="/wiki/Giusto_Bellavitis" title="Giusto Bellavitis">Giusto Bellavitis</a> de bipoint, que es un segmento orientado, uno de cuyos extremos es el origen y el otro un objetivo. Los vectores se reconsideraron con la presentación de los <a href="/wiki/N%C3%BAmeros_complejos" class="mw-redirect" title="Números complejos">números complejos</a> de <a href="/wiki/Jean-Robert_Argand" title="Jean-Robert Argand">Argand</a> y <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a> y la creación de los <a href="/wiki/Cuaterni%C3%B3n" title="Cuaternión">cuaterniones</a> por este último (Hamilton fue además el que inventó el nombre de vector).<sup id="cite_ref-4" class="reference separada"><a href="#cite_note-4"><span class="corchete-llamada">[</span>nota 4<span class="corchete-llamada">]</span></a></sup> Son elementos de <b>R</b><sup>2</sup> y <b>R</b><sup>4</sup>; el tratamiento mediante <a href="/wiki/Combinaciones_lineales" class="mw-redirect" title="Combinaciones lineales">combinaciones lineales</a> se remonta a <a href="/wiki/Laguerre" class="mw-redirect" title="Laguerre">Laguerre</a> en 1867, quien también definió los <a href="/wiki/Sistemas_de_ecuaciones_lineales" class="mw-redirect" title="Sistemas de ecuaciones lineales">sistemas de ecuaciones lineales</a>. </p><p>En 1857, <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Cayley</a> introdujo la <a href="/wiki/Notaci%C3%B3n_matricial" class="mw-redirect" title="Notación matricial">notación matricial</a> que permite una armonización y simplificación de las <a href="/wiki/Aplicaciones_lineales" class="mw-redirect" title="Aplicaciones lineales">aplicaciones lineales</a>. Casi al mismo tiempo, <a href="/wiki/Grassmann" class="mw-redirect" title="Grassmann">Grassmann</a> estudió el cálculo baricéntrico iniciado por Möbius. Previó conjuntos de objetos abstractos dotados de operaciones.<sup id="cite_ref-5" class="reference separada"><a href="#cite_note-5"><span class="corchete-llamada">[</span>nota 5<span class="corchete-llamada">]</span></a></sup> En su trabajo, los conceptos de <a href="/wiki/Independencia_lineal" class="mw-redirect" title="Independencia lineal">independencia lineal</a> y <a href="/wiki/Dimensi%C3%B3n" title="Dimensión">dimensión</a>, así como de <a href="/wiki/Producto_escalar" title="Producto escalar">producto escalar</a> están presentes. En realidad el trabajo de Grassmann de 1844 supera el marco de los espacios vectoriales, ya que teniendo en cuenta la multiplicación, también, lo llevó a lo que hoy en día se llaman <a href="/wiki/%C3%81lgebra" title="Álgebra">álgebras</a>. El matemático italiano <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Peano</a> dio la primera definición moderna de espacios vectoriales y aplicaciones lineales en 1888.<sup id="cite_ref-6" class="reference separada"><a href="#cite_note-6"><span class="corchete-llamada">[</span>nota 6<span class="corchete-llamada">]</span></a></sup> </p><p>Un desarrollo importante de los espacios vectoriales se debe a la construcción de los <a href="/wiki/Espacios_de_funciones" class="mw-redirect" title="Espacios de funciones">espacios de funciones</a> por <a href="/wiki/Henri_Lebesgue" class="mw-redirect" title="Henri Lebesgue">Henri Lebesgue</a>. Esto más tarde fue formalizado por <a href="/wiki/Stefan_Banach" title="Stefan Banach">Banach</a> en su tesis doctoral de 1920<sup id="cite_ref-7" class="reference separada"><a href="#cite_note-7"><span class="corchete-llamada">[</span>nota 7<span class="corchete-llamada">]</span></a></sup> y por <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a>. En este momento, el <a href="/wiki/%C3%81lgebra" title="Álgebra">álgebra</a> y el nuevo campo del <a href="/wiki/An%C3%A1lisis_funcional" title="Análisis funcional">análisis funcional</a> empezaron a interactuar, en particular con conceptos clave tales como los <a href="/wiki/Espacios_Lp" title="Espacios Lp">espacios de funciones p-integrables</a> y los <a href="/wiki/Espacios_de_Hilbert" class="mw-redirect" title="Espacios de Hilbert">espacios de Hilbert</a>. También en este tiempo, los primeros estudios sobre espacios vectoriales de infinitas dimensiones se realizaron. </p><p>Los espacios vectoriales tienen aplicaciones en otras ramas de la matemática, la <a href="/wiki/Ciencia" title="Ciencia">ciencia</a> y la <a href="/wiki/Ingenier%C3%ADa" title="Ingeniería">ingeniería</a>. Se utilizan en métodos como las <a href="/wiki/Series_de_Fourier" class="mw-redirect" title="Series de Fourier">series de Fourier</a>, que se utiliza en las rutinas modernas de <a href="/wiki/Compresi%C3%B3n_de_im%C3%A1genes" class="mw-redirect" title="Compresión de imágenes">compresión de imágenes</a> y sonido, o proporcionan el marco para resolver <a href="/wiki/Ecuaciones_en_derivadas_parciales" class="mw-redirect" title="Ecuaciones en derivadas parciales">ecuaciones en derivadas parciales</a>. Además, los espacios vectoriales proporcionan una forma abstracta libre de coordenadas de tratar con objetos geométricos y físicos, tales como <a href="/wiki/Tensor" title="Tensor">tensores</a>, que a su vez permiten estudiar las propiedades locales de <a href="/wiki/Variedad_(matem%C3%A1tica)" class="mw-redirect" title="Variedad (matemática)">variedades</a> mediante técnicas de linealización. </p> <div class="mw-heading mw-heading2"><h2 id="Notación"><span id="Notaci.C3.B3n"></span>Notación</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=2" title="Editar sección: Notación"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dado un espacio vectorial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> sobre un cuerpo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>, se distinguen los elementos de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> y los de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>. </p><p>Los elementos de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> suelen denotarse por </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} ,\mathbf {v} ,\mathbf {w} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} ,\mathbf {v} ,\mathbf {w} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63824714d995834ae5070ccfc9c85ad6cd44082d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.895ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} ,\mathbf {v} ,\mathbf {w} }"></span></dd></dl> <p>y son llamados <b>vectores</b>. </p><p>Dependiendo las fuentes que se consulten, también es común denotarlos por </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {u}},{\bar {v}},{\bar {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {u}},{\bar {v}},{\bar {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dd57b4168b611e5c7133b38ce3d13a4e24a3dfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.289ex; height:2.343ex;" alt="{\displaystyle {\bar {u}},{\bar {v}},{\bar {w}}}"></span></dd></dl> <p>y si el texto es de física entonces suelen denotarse por </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}},{\vec {v}},{\vec {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}},{\vec {v}},{\vec {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/302f55f1883a735761a86e46d8ba6f24a8841b77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.676ex;" alt="{\displaystyle {\vec {u}},{\vec {v}},{\vec {w}}}"></span></dd></dl> <p>Mientras que los elementos de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> se denotan como </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,\alpha ,\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,\alpha ,\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac69e6b9cc753a595228308a53e02f57685a536f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.149ex; height:2.509ex;" alt="{\displaystyle a,b,\alpha ,\beta }"></span></dd></dl> <p>y son llamados <b>escalares</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Definición"><span id="Definici.C3.B3n"></span>Definición</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=3" title="Editar sección: Definición"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un <b>espacio vectorial</b> sobre un <a href="/wiki/Cuerpo_(matem%C3%A1tica)" class="mw-redirect" title="Cuerpo (matemática)">cuerpo</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> (como el cuerpo de los <a href="/wiki/N%C3%BAmero_real" title="Número real">números reales</a> o los <a href="/wiki/N%C3%BAmero_complejo" title="Número complejo">números complejos</a>) es un <a href="/wiki/Conjunto" title="Conjunto">conjunto</a> no vacío, digamos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, dotado de dos operaciones para las cuales será cerrado: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{llccl}{\mbox{Suma}}&+:&{V\times V}&\rightarrow &{V}\\&&(\mathbf {u} ,\mathbf {v} )&\mapsto &\mathbf {u} +\mathbf {v} \end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left left center center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Suma</mtext> </mstyle> </mrow> </mtd> <mtd> <mo>+</mo> <mo>:</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> <mo>×</mo> <mi>V</mi> </mrow> </mtd> <mtd> <mo stretchy="false">→</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mo stretchy="false">↦</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{llccl}{\mbox{Suma}}&+:&{V\times V}&\rightarrow &{V}\\&&(\mathbf {u} ,\mathbf {v} )&\mapsto &\mathbf {u} +\mathbf {v} \end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fecee574bace502d0ca9a7ab0fbd54a333176d6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.301ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{llccl}{\mbox{Suma}}&+:&{V\times V}&\rightarrow &{V}\\&&(\mathbf {u} ,\mathbf {v} )&\mapsto &\mathbf {u} +\mathbf {v} \end{array}}}"></span></dd></dl> <p>operación interna tal que: </p> <ul><li>Tenga la <a href="/wiki/Propiedad_conmutativa" class="mw-redirect" title="Propiedad conmutativa">propiedad conmutativa</a>:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} +\mathbf {v} =\mathbf {v} +\mathbf {u} ,\quad \forall \;\mathbf {u} ,\mathbf {v} \in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} +\mathbf {v} =\mathbf {v} +\mathbf {u} ,\quad \forall \;\mathbf {u} ,\mathbf {v} \in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db040c69428368fa4a05b04d008f05a26a530ce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.424ex; height:2.509ex;" alt="{\displaystyle \mathbf {u} +\mathbf {v} =\mathbf {v} +\mathbf {u} ,\quad \forall \;\mathbf {u} ,\mathbf {v} \in V}"></span></dd></dl> <ul><li>Tenga la <a href="/wiki/Propiedad_asociativa" class="mw-redirect" title="Propiedad asociativa">propiedad asociativa</a>:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} +(\mathbf {v} +\mathbf {w} )=(\mathbf {u} +\mathbf {v} )+\mathbf {w} ,\quad \forall \;\mathbf {u} ,\mathbf {v} ,\mathbf {w} \in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} +(\mathbf {v} +\mathbf {w} )=(\mathbf {u} +\mathbf {v} )+\mathbf {w} ,\quad \forall \;\mathbf {u} ,\mathbf {v} ,\mathbf {w} \in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b897ff1debc928701150ba752eb31fd437609c25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.551ex; height:2.843ex;" alt="{\displaystyle \mathbf {u} +(\mathbf {v} +\mathbf {w} )=(\mathbf {u} +\mathbf {v} )+\mathbf {w} ,\quad \forall \;\mathbf {u} ,\mathbf {v} ,\mathbf {w} \in V}"></span></dd></dl> <ul><li>Exista el <a href="/wiki/Elemento_neutro" title="Elemento neutro">elemento neutro</a>:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists \;\mathbf {e} \in {}V:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>V</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists \;\mathbf {e} \in {}V:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2477bfd8fa4c9fe71a572de31216a901dac0dbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.083ex; height:2.176ex;" alt="{\displaystyle \exists \;\mathbf {e} \in {}V:}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} +\mathbf {e} =\mathbf {u} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} +\mathbf {e} =\mathbf {u} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67219eebd3731c4fefcc663876d4cf6924ade5cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.781ex; height:2.343ex;" alt="{\displaystyle \mathbf {u} +\mathbf {e} =\mathbf {u} ,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \;\mathbf {u} \in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \;\mathbf {u} \in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1482a6a0c2e9952027810eab134aef9f8b02ce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.051ex; height:2.176ex;" alt="{\displaystyle \forall \;\mathbf {u} \in V}"></span></dd></dl> <ul><li>Exista el <a href="/wiki/Elemento_opuesto" class="mw-redirect" title="Elemento opuesto">elemento opuesto</a>:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \;\mathbf {u} \in V,\quad \exists -\mathbf {u} \in {}V:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>∈</mo> <mi>V</mi> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∃</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>V</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \;\mathbf {u} \in V,\quad \exists -\mathbf {u} \in {}V:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b71f8b84512e75f7676847242777a3f21d3360a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.946ex; height:2.509ex;" alt="{\displaystyle \forall \;\mathbf {u} \in V,\quad \exists -\mathbf {u} \in {}V:}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} +(-\mathbf {u} )=\mathbf {e} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} +(-\mathbf {u} )=\mathbf {e} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9a0740293206ba06f1a4f181822c3cd27873f6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.752ex; height:2.843ex;" alt="{\displaystyle \mathbf {u} +(-\mathbf {u} )=\mathbf {e} }"></span></dd></dl> <p>Y tenga la operación producto por un escalar: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{llccl}{\mbox{Producto}}&\cdot {}:&{K\times V}&\rightarrow &{V}\\&&{(a,\mathbf {u} )}&\mapsto &a\cdot \mathbf {u} \end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left left center center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Producto</mtext> </mstyle> </mrow> </mtd> <mtd> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>:</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mo>×</mo> <mi>V</mi> </mrow> </mtd> <mtd> <mo stretchy="false">→</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </mtd> <mtd> <mo stretchy="false">↦</mo> </mtd> <mtd> <mi>a</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{llccl}{\mbox{Producto}}&\cdot {}:&{K\times V}&\rightarrow &{V}\\&&{(a,\mathbf {u} )}&\mapsto &a\cdot \mathbf {u} \end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7f04a7465bbfe37cde2812d32f3ddabf62ec241" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.734ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{llccl}{\mbox{Producto}}&\cdot {}:&{K\times V}&\rightarrow &{V}\\&&{(a,\mathbf {u} )}&\mapsto &a\cdot \mathbf {u} \end{array}}}"></span></dd></dl> <p>operación externa tal que: </p> <ul><li>Tenga la <a href="/wiki/Propiedad_asociativa" class="mw-redirect" title="Propiedad asociativa">propiedad asociativa</a>:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot ({\mathit {b}}\cdot \mathbf {u} )=({\mathit {a}}\cdot {\mathit {b}})\cdot \mathbf {u} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot ({\mathit {b}}\cdot \mathbf {u} )=({\mathit {a}}\cdot {\mathit {b}})\cdot \mathbf {u} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f129e81105c4fa0723a9d822ea87ad9b38eddfec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.566ex; height:2.843ex;" alt="{\displaystyle {\mathit {a}}\cdot ({\mathit {b}}\cdot \mathbf {u} )=({\mathit {a}}\cdot {\mathit {b}})\cdot \mathbf {u} ,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \;{\mathit {a}},{\mathit {b}}\in {}K,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>K</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \;{\mathit {a}},{\mathit {b}}\in {}K,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/903af49eedb5c86ff893be7cd504deebbfeea03c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.783ex; height:2.509ex;" alt="{\displaystyle \forall \;{\mathit {a}},{\mathit {b}}\in {}K,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \;\mathbf {u} \in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \;\mathbf {u} \in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1482a6a0c2e9952027810eab134aef9f8b02ce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.051ex; height:2.176ex;" alt="{\displaystyle \forall \;\mathbf {u} \in V}"></span></dd></dl> <ul><li>Exista el <a href="/wiki/Elemento_neutro" title="Elemento neutro">elemento neutro</a>:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists e\in {K}:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>e</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists e\in {K}:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac1ae7f935db4795fbb9df1131f44af8d2d967c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.575ex; height:2.176ex;" alt="{\displaystyle \exists e\in {K}:}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\cdot \mathbf {u} =\mathbf {u} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\cdot \mathbf {u} =\mathbf {u} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ca125eaca0a6c2db0b59d8e440eacf1e4f23ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.478ex; height:2.009ex;" alt="{\displaystyle e\cdot \mathbf {u} =\mathbf {u} ,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \;\mathbf {u} \in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \;\mathbf {u} \in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1482a6a0c2e9952027810eab134aef9f8b02ce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.051ex; height:2.176ex;" alt="{\displaystyle \forall \;\mathbf {u} \in V}"></span></dd></dl> <ul><li>Tenga la <a href="/wiki/Propiedad_distributiva" class="mw-redirect" title="Propiedad distributiva">propiedad distributiva</a> respecto de la suma vectorial:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot (\mathbf {u} +\mathbf {v} )={\mathit {a}}\cdot \mathbf {u} +{\mathit {a}}\cdot \mathbf {v} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot (\mathbf {u} +\mathbf {v} )={\mathit {a}}\cdot \mathbf {u} +{\mathit {a}}\cdot \mathbf {v} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58905cf0beb5f3ea4dccbd238aecf17745f9142e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.629ex; height:2.843ex;" alt="{\displaystyle {\mathit {a}}\cdot (\mathbf {u} +\mathbf {v} )={\mathit {a}}\cdot \mathbf {u} +{\mathit {a}}\cdot \mathbf {v} ,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \;{\mathit {a}}\in K,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>∈</mo> <mi>K</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \;{\mathit {a}}\in K,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c0911331bc7de7637f8a7742a2e623048209357" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.679ex; height:2.509ex;" alt="{\displaystyle \forall \;{\mathit {a}}\in K,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \;\mathbf {u} ,\mathbf {v} \in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \;\mathbf {u} ,\mathbf {v} \in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b2030bd06073ed107c1cb9696c1a21b65c95ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.496ex; height:2.509ex;" alt="{\displaystyle \forall \;\mathbf {u} ,\mathbf {v} \in V}"></span></dd></dl> <ul><li>Tenga la <a href="/wiki/Propiedad_distributiva" class="mw-redirect" title="Propiedad distributiva">propiedad distributiva</a> respecto de la suma escalar:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot \mathbf {u} ={\mathit {a}}\cdot \mathbf {u} +{\mathit {b}}\cdot \mathbf {u} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot \mathbf {u} ={\mathit {a}}\cdot \mathbf {u} +{\mathit {b}}\cdot \mathbf {u} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f207a31049328f8eef2bfcfe937482ecfd1fcea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.244ex; height:2.843ex;" alt="{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot \mathbf {u} ={\mathit {a}}\cdot \mathbf {u} +{\mathit {b}}\cdot \mathbf {u} ,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall {}{\mathit {a}},{\mathit {b}}\in {}K,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>K</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall {}{\mathit {a}},{\mathit {b}}\in {}K,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d48e6c3f0a4ecf4af7f02f8a11ad428919e5792" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.137ex; height:2.509ex;" alt="{\displaystyle \forall {}{\mathit {a}},{\mathit {b}}\in {}K,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \;\mathbf {u} \in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \;\mathbf {u} \in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1482a6a0c2e9952027810eab134aef9f8b02ce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.051ex; height:2.176ex;" alt="{\displaystyle \forall \;\mathbf {u} \in V}"></span></dd></dl> <div class="VT rellink"><span style="font-size:88%">Véase también:</span> <i><a href="/wiki/Espacio_eucl%C3%ADdeo" title="Espacio euclídeo">Espacio euclídeo</a></i></div> <div class="VT rellink"><span style="font-size:88%">Véase también:</span> <i><a href="/wiki/Vector" title="Vector">Vector</a></i></div> <div class="VT rellink"><span style="font-size:88%">Véase también:</span> <i><a href="/wiki/Vector#Representación_gráfica_de_vectores" title="Vector">Representación gráfica de vectores</a></i></div> <div class="mw-heading mw-heading3"><h3 id="Observaciones">Observaciones</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=4" title="Editar sección: Observaciones"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La denominación de las dos operaciones no condiciona la definición de espacio vectorial por lo que es habitual encontrar traducciones de obras en las que se utiliza <i>multiplicación</i> para el <i>producto</i> y <i>adición</i> para la <i>suma</i>, usando las distinciones propias de la aritmética. </p><p>Para demostrar que un conjunto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9501977c158ec48ec5ee3aa83d13978fa553ec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V_{}^{}}"></span> es un espacio vectorial: </p> <ul><li>Lo es si sus dos operaciones, por ejemplo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \odot (V,V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊙</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \odot (V,V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6215fd3e0cc73177f2c99c8c8dcc09152ad5adcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.226ex; height:2.843ex;" alt="{\displaystyle \odot (V,V)}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ast (V,K),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ast (V,K),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e1983bcd36088f41deaee00f4b939a8a77ab5ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.506ex; height:2.843ex;" alt="{\displaystyle \ast (V,K),}"></span> admiten una redefinición del tipo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +(V,V)=\odot (V,V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>⊙</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +(V,V)=\odot (V,V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/508b0e2d1d0c6325537d54d93afb1ba493c49899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.55ex; height:2.843ex;" alt="{\displaystyle +(V,V)=\odot (V,V)}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot (K,V)=\ast (V,K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot (K,V)=\ast (V,K)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/192144d16e9fbd3139a643427e001a6e55872a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.301ex; height:2.843ex;" alt="{\displaystyle \cdot (K,V)=\ast (V,K)}"></span> cumpliendo las 8 condiciones exigidas.</li> <li>Si supiésemos que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9501977c158ec48ec5ee3aa83d13978fa553ec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V_{}^{}}"></span> es un <a href="/wiki/Grupo_conmutativo" class="mw-redirect" title="Grupo conmutativo">grupo conmutativo</a> o abeliano respecto la suma ya tendríamos probados los apartados <b>1, 2, 3</b> y <b>4</b>.</li> <li>Si supiésemos que el producto es una <a href="/wiki/Acci%C3%B3n_(matem%C3%A1tica)" title="Acción (matemática)">acción</a> por la izquierda de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9501977c158ec48ec5ee3aa83d13978fa553ec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V_{}^{}}"></span> tendríamos probados los apartados <b>5</b> y <b>6</b>.</li> <li>Si no, se dice lo contrario:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\mathbf {v} \neq \mathbf {v} {\mathit {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\mathbf {v} \neq \mathbf {v} {\mathit {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77906507e32bf4c8de5609f18aa47133cac3bfff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.074ex; width:8.371ex; height:2.676ex;" alt="{\displaystyle {\mathit {a}}\mathbf {v} \neq \mathbf {v} {\mathit {a}}}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Propiedades">Propiedades</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=5" title="Editar sección: Propiedades"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="mw-collapsible wikitable mw-collapsed" width="75%" style="text-align:left; padding:0px; background-color: var(--background-color-neutral-subtle, #f9f9f9); color: var(--color-base, #202122);"> <tbody><tr> <td style="text-align:left; font-weight:bold;">Unicidad del vector neutro de la propiedad <b>3</b> </td></tr> <tr> <td style="background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align:left; font-size:95%; padding:6px">supongamos que el neutro no es único, es decir, sean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0_{1}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn mathvariant="bold">0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0_{1}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35d38c4296a0446db4302893c2946c6992f5f2cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.514ex; height:2.509ex;" alt="{\displaystyle \mathbf {0_{1}} }"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0_{2}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn mathvariant="bold">0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0_{2}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba0eb3f547effe3bacaf9d8b7341869db5a4887" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.514ex; height:2.509ex;" alt="{\displaystyle \mathbf {0_{2}} }"></span> dos vectores neutros, entonces: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\begin{array}{l}\mathbf {u} +\mathbf {0_{1}} =\mathbf {u} \\\mathbf {u} +\mathbf {0_{2}} =\mathbf {u} \end{array}}\right\}\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn mathvariant="bold">0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn mathvariant="bold">0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\begin{array}{l}\mathbf {u} +\mathbf {0_{1}} =\mathbf {u} \\\mathbf {u} +\mathbf {0_{2}} =\mathbf {u} \end{array}}\right\}\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0272ddc848fceb222506d3e0df0600d490f22fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.887ex; height:6.176ex;" alt="{\displaystyle \left.{\begin{array}{l}\mathbf {u} +\mathbf {0_{1}} =\mathbf {u} \\\mathbf {u} +\mathbf {0_{2}} =\mathbf {u} \end{array}}\right\}\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} +\mathbf {0_{1}} =\mathbf {u} +\mathbf {0_{2}} \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn mathvariant="bold">0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn mathvariant="bold">0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} +\mathbf {0_{1}} =\mathbf {u} +\mathbf {0_{2}} \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8531d82d892dd3914f0074088e7df23c41f5b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.747ex; height:2.509ex;" alt="{\displaystyle \mathbf {u} +\mathbf {0_{1}} =\mathbf {u} +\mathbf {0_{2}} \Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0_{1}} =\mathbf {0_{2}} \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn mathvariant="bold">0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn mathvariant="bold">0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0_{1}} =\mathbf {0_{2}} \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc00099f84b323cd02f29167fb60f76b49fecab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.096ex; height:2.509ex;" alt="{\displaystyle \mathbf {0_{1}} =\mathbf {0_{2}} \Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists !\;\mathbf {0} \in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mo>!</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists !\;\mathbf {0} \in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa2b3048decdb664aeab9f8642c3d5b64f960530" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.549ex; height:2.176ex;" alt="{\displaystyle \exists !\;\mathbf {0} \in V}"></span></dd></dl> </td></tr></tbody></table> <table class="mw-collapsible wikitable mw-collapsed" width="75%" style="text-align:left; padding:0px; background-color: var(--background-color-neutral-subtle, #f9f9f9); color: var(--color-base, #202122);"> <tbody><tr> <td style="text-align:left; font-weight:bold;">Unicidad del vector opuesto de la propiedad <b>4</b> </td></tr> <tr> <td style="background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align:left; font-size:95%; padding:6px">supongamos que el opuesto no es único, es decir, sean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {-u_{1}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold">−</mo> <msub> <mi mathvariant="bold">u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {-u_{1}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bfd4530b080168f9ee4466a0dfc41c412aa4f1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.74ex; height:2.009ex;" alt="{\displaystyle \mathbf {-u_{1}} }"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {-u_{2}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold">−</mo> <msub> <mi mathvariant="bold">u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {-u_{2}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad75ef93b6b56588ab0f6a61b3439bd623bd2e8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.74ex; height:2.009ex;" alt="{\displaystyle \mathbf {-u_{2}} }"></span> dos vectores opuestos de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }"></span>, entonces, como el neutro es único: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\begin{array}{l}\mathbf {u} -\mathbf {u_{1}} =\mathbf {0} \\\mathbf {u} -\mathbf {u_{2}} =\mathbf {0} \end{array}}\right\}\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\begin{array}{l}\mathbf {u} -\mathbf {u_{1}} =\mathbf {0} \\\mathbf {u} -\mathbf {u_{2}} =\mathbf {0} \end{array}}\right\}\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a9c87aae7eb83883b1efe46c7e2532e01744467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.887ex; height:6.176ex;" alt="{\displaystyle \left.{\begin{array}{l}\mathbf {u} -\mathbf {u_{1}} =\mathbf {0} \\\mathbf {u} -\mathbf {u_{2}} =\mathbf {0} \end{array}}\right\}\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} -\mathbf {u_{1}} =\mathbf {u} -\mathbf {u_{2}} \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} -\mathbf {u_{1}} =\mathbf {u} -\mathbf {u_{2}} \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fdc4379c43ea6e0cff14626110c78d411ba56e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.044ex; height:2.343ex;" alt="{\displaystyle \mathbf {u} -\mathbf {u_{1}} =\mathbf {u} -\mathbf {u_{2}} \Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\mathbf {u_{1}} =-\mathbf {u_{2}} \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\mathbf {u_{1}} =-\mathbf {u_{2}} \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98ec84869634b45db5f29b6d41737d6cd74561db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.009ex; height:2.343ex;" alt="{\displaystyle -\mathbf {u_{1}} =-\mathbf {u_{2}} \Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists !-\mathbf {u} \in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mo>!</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists !-\mathbf {u} \in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dcba3254b9898ef023f0072340b849f0cfbbacb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.893ex; height:2.343ex;" alt="{\displaystyle \exists !-\mathbf {u} \in V}"></span></dd></dl> </td></tr></tbody></table> <table class="mw-collapsible wikitable mw-collapsed" width="75%" style="text-align:left; padding:0px; background-color: var(--background-color-neutral-subtle, #f9f9f9); color: var(--color-base, #202122);"> <tbody><tr> <td style="text-align:left; font-weight:bold;">Unicidad del elemento <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b175ad4cc2ea14b4b80581484553042ced2525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1_{}^{}}"></span> en el cuerpo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a6c0f78592eaf8c5b477d9551ea3adffeea33fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K_{}^{}}"></span> </td></tr> <tr> <td style="background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align:left; font-size:95%; padding:6px">supongamos que 1 no es único, es decir, sean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {1_{1}}}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </msub> </mrow> </mrow> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {1_{1}}}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/346214551a9c3feb6e72d41854c64a6ae0d1b2fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.905ex; height:2.509ex;" alt="{\displaystyle {\mathit {1_{1}}}\;}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {1_{2}}}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">2</mn> </mrow> </msub> </mrow> </mrow> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {1_{2}}}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab02cd1a2af98a4f9cdf601874933d81ac0410b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.905ex; height:2.509ex;" alt="{\displaystyle {\mathit {1_{2}}}\;}"></span> dos unidades, entonces: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\begin{array}{l}{\mathit {a}}\cdot {\mathit {1_{1}}}={\mathit {a}}\\{\mathit {a}}\cdot {\mathit {1_{2}}}={\mathit {a}}\end{array}}\right\}\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </msub> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">2</mn> </mrow> </msub> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\begin{array}{l}{\mathit {a}}\cdot {\mathit {1_{1}}}={\mathit {a}}\\{\mathit {a}}\cdot {\mathit {1_{2}}}={\mathit {a}}\end{array}}\right\}\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1091b2469db04dcf7bb28606151d239648816a9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.877ex; height:6.176ex;" alt="{\displaystyle \left.{\begin{array}{l}{\mathit {a}}\cdot {\mathit {1_{1}}}={\mathit {a}}\\{\mathit {a}}\cdot {\mathit {1_{2}}}={\mathit {a}}\end{array}}\right\}\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot {\mathit {1_{1}}}={\mathit {a}}\cdot {\mathit {1_{2}}}\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </msub> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">2</mn> </mrow> </msub> </mrow> </mrow> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot {\mathit {1_{1}}}={\mathit {a}}\cdot {\mathit {1_{2}}}\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13155e0e889d3eed3e63d74d93294e5a91229a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.322ex; height:2.509ex;" alt="{\displaystyle {\mathit {a}}\cdot {\mathit {1_{1}}}={\mathit {a}}\cdot {\mathit {1_{2}}}\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {1_{1}}}={\mathit {1_{2}}}\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </msub> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">2</mn> </mrow> </msub> </mrow> </mrow> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {1_{1}}}={\mathit {1_{2}}}\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bf9cb92ff9d2cc600529c1098825ac1e18b44e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.588ex; height:2.509ex;" alt="{\displaystyle {\mathit {1_{1}}}={\mathit {1_{2}}}\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists !\;{\mathit {1}}\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mo>!</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </mrow> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists !\;{\mathit {1}}\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0fec78c079e1161d976200960926e7f4c410c56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.679ex; height:2.176ex;" alt="{\displaystyle \exists !\;{\mathit {1}}\in K}"></span></dd></dl> </td></tr></tbody></table> <table class="mw-collapsible wikitable mw-collapsed" width="75%" style="text-align:left; padding:0px; background-color: var(--background-color-neutral-subtle, #f9f9f9); color: var(--color-base, #202122);"> <tbody><tr> <td style="text-align:left; font-weight:bold;">Unicidad del elemento inverso en el cuerpo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a6c0f78592eaf8c5b477d9551ea3adffeea33fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K_{}^{}}"></span> </td></tr> <tr> <td style="background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align:left; font-size:95%; padding:6px">supongamos que el inverso <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{}^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dbb84236310c25ee33d71852cff7096a7b8ed4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.563ex; height:2.676ex;" alt="{\displaystyle a_{}^{-1}}"></span> de a, no es único, es decir, sean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eac43c15969120a96d205fd5beee9ff578edda5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.563ex; height:3.343ex;" alt="{\displaystyle a_{1}^{-1}}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{2}^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e524445fdd1c9476cfbdbe87a5d43ca39210abc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.563ex; height:3.343ex;" alt="{\displaystyle a_{2}^{-1}}"></span> dos opuestos de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a40555722f80369dc9cbe8266ae8706a45f0010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a_{}^{}}"></span>, entonces, como el neutro es único: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\begin{array}{l}{\mathit {a}}\cdot {\mathit {a_{1}^{-1}}}={\mathit {1}}\\{\mathit {a}}\cdot {\mathit {a_{2}^{-1}}}={\mathit {1}}\end{array}}\right\}\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo class="MJX-tex-mathit" mathvariant="italic">−</mo> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </msubsup> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo class="MJX-tex-mathit" mathvariant="italic">−</mo> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </msubsup> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\begin{array}{l}{\mathit {a}}\cdot {\mathit {a_{1}^{-1}}}={\mathit {1}}\\{\mathit {a}}\cdot {\mathit {a_{2}^{-1}}}={\mathit {1}}\end{array}}\right\}\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f016f485f650748ba0921df47a91b16b63656b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.286ex; height:7.509ex;" alt="{\displaystyle \left.{\begin{array}{l}{\mathit {a}}\cdot {\mathit {a_{1}^{-1}}}={\mathit {1}}\\{\mathit {a}}\cdot {\mathit {a_{2}^{-1}}}={\mathit {1}}\end{array}}\right\}\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot {\mathit {a_{1}^{-1}}}={\mathit {a}}\cdot {\mathit {a_{2}^{-1}}}\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo class="MJX-tex-mathit" mathvariant="italic">−</mo> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </msubsup> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo class="MJX-tex-mathit" mathvariant="italic">−</mo> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </msubsup> </mrow> </mrow> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot {\mathit {a_{1}^{-1}}}={\mathit {a}}\cdot {\mathit {a_{2}^{-1}}}\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5347dbde6380cd5c5723f51ab8367476f4e61727" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.879ex; height:3.343ex;" alt="{\displaystyle {\mathit {a}}\cdot {\mathit {a_{1}^{-1}}}={\mathit {a}}\cdot {\mathit {a_{2}^{-1}}}\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a_{1}^{-1}}}={\mathit {a_{2}^{-1}}}\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo class="MJX-tex-mathit" mathvariant="italic">−</mo> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </msubsup> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo class="MJX-tex-mathit" mathvariant="italic">−</mo> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </msubsup> </mrow> </mrow> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a_{1}^{-1}}}={\mathit {a_{2}^{-1}}}\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d71813d1c302387b99a2d635c2fae39e7dd178e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.145ex; height:3.343ex;" alt="{\displaystyle {\mathit {a_{1}^{-1}}}={\mathit {a_{2}^{-1}}}\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists !{\mathit {a^{-1}}}\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo class="MJX-tex-mathit" mathvariant="italic">−</mo> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </msup> </mrow> </mrow> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists !{\mathit {a^{-1}}}\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e02790bf43355c4da2941a2c3741d2ce20e7fa50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.385ex; height:2.676ex;" alt="{\displaystyle \exists !{\mathit {a^{-1}}}\in K}"></span></dd></dl> </td></tr></tbody></table> <table class="mw-collapsible wikitable mw-collapsed" width="75%" style="text-align:left; padding:0px; background-color: var(--background-color-neutral-subtle, #f9f9f9); color: var(--color-base, #202122);"> <tbody><tr> <td style="text-align:left; font-weight:bold;">Producto de un escalar por el vector neutro </td></tr> <tr> <td style="background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align:left; font-size:95%; padding:6px"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot \mathbf {u} =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot \mathbf {u} =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba02648817c760a6e49f7ff46ef0951e3b34c163" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.806ex; height:1.676ex;" alt="{\displaystyle {\mathit {a}}\cdot \mathbf {u} =}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot (\mathbf {u} +\mathbf {0} )=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot (\mathbf {u} +\mathbf {0} )=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6504670cd32fce70a461dd48b9f3661473472e2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.792ex; height:2.843ex;" alt="{\displaystyle {\mathit {a}}\cdot (\mathbf {u} +\mathbf {0} )=}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot \mathbf {u} +{\mathit {a}}\cdot \mathbf {0} \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot \mathbf {u} +{\mathit {a}}\cdot \mathbf {0} \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bb7b63fca1c5cd103c3b46ce663e8122dee1a26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.365ex; height:2.343ex;" alt="{\displaystyle {\mathit {a}}\cdot \mathbf {u} +{\mathit {a}}\cdot \mathbf {0} \Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot \mathbf {0} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot \mathbf {0} =\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f859e5455a9fe976660349ccf612b314e14bea2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.639ex; height:2.176ex;" alt="{\displaystyle {\mathit {a}}\cdot \mathbf {0} =\mathbf {0} }"></span> </td></tr></tbody></table> <table class="mw-collapsible wikitable mw-collapsed" width="75%" style="text-align:left; padding:0px; background-color: var(--background-color-neutral-subtle, #f9f9f9); color: var(--color-base, #202122);"> <tbody><tr> <td style="text-align:left; font-weight:bold;">Producto del escalar 0 por un vector </td></tr> <tr> <td style="background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align:left; font-size:95%; padding:6px"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95be4f52cc98700c44d215c13d1290dd5cad1a57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.939ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} =}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {1}}\cdot \mathbf {u} =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {1}}\cdot \mathbf {u} =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04294bc56135ffbacae3625cf8a2d7d5dbb41de6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.806ex; height:2.176ex;" alt="{\displaystyle {\mathit {1}}\cdot \mathbf {u} =}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathit {1}}+{\mathit {0}})\cdot \mathbf {u} =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">0</mn> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathit {1}}+{\mathit {0}})\cdot \mathbf {u} =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27e92748ab370d6f256e554d44334098b561dac7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.643ex; height:2.843ex;" alt="{\displaystyle ({\mathit {1}}+{\mathit {0}})\cdot \mathbf {u} =}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {1}}\cdot \mathbf {u} +{\mathit {0}}\cdot \mathbf {u} =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">0</mn> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {1}}\cdot \mathbf {u} +{\mathit {0}}\cdot \mathbf {u} =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff89f7ebc2b29b773962fe6960992f402df7f95e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.999ex; height:2.343ex;" alt="{\displaystyle {\mathit {1}}\cdot \mathbf {u} +{\mathit {0}}\cdot \mathbf {u} =}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} +{\mathit {0}}\cdot \mathbf {u} \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">0</mn> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} +{\mathit {0}}\cdot \mathbf {u} \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77a174bdc7412c79e3b67bbfc9ba4da04322f4c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.647ex; height:2.343ex;" alt="{\displaystyle \mathbf {u} +{\mathit {0}}\cdot \mathbf {u} \Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {0}}\cdot \mathbf {u} =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">0</mn> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {0}}\cdot \mathbf {u} =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda1e4b92cd44cfd27f2120faf39eb161c197ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.806ex; height:2.176ex;" alt="{\displaystyle {\mathit {0}}\cdot \mathbf {u} =}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} }"></span> </td></tr></tbody></table> <p>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot \mathbf {u} =\mathbf {0} \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot \mathbf {u} =\mathbf {0} \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e62d4b68c16639a6777df748fe05f6a62ca3614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.756ex; height:2.176ex;" alt="{\displaystyle {\mathit {a}}\cdot \mathbf {u} =\mathbf {0} \Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}={\mathit {0}}\quad \lor \quad \mathbf {u} =\mathbf {0} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">0</mn> </mrow> </mrow> <mspace width="1em" /> <mo>∨</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}={\mathit {0}}\quad \lor \quad \mathbf {u} =\mathbf {0} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4706f8f131a6c8fc1a125b03725aedd0c7cf8412" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.269ex; height:2.176ex;" alt="{\displaystyle {\mathit {a}}={\mathit {0}}\quad \lor \quad \mathbf {u} =\mathbf {0} .}"></span> </p> <ul><li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{}^{}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{}^{}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13afbf8298bf0a8b36b16d8130cba2a0c3866966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.138ex; height:2.509ex;" alt="{\displaystyle a_{}^{}=0,}"></span> es cierto.</li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2068d811f09c9ca1437234192c052786eae4e41f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.138ex; height:2.676ex;" alt="{\displaystyle a\neq 0,}"></span> entonces:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists !\;a^{-1}\in K:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mo>!</mo> <mspace width="thickmathspace" /> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>∈</mo> <mi>K</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists !\;a^{-1}\in K:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d085fde6d965e80caa1976a9bd6b533e0210bd3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.346ex; height:2.676ex;" alt="{\displaystyle \exists !\;a^{-1}\in K:}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{-1}a=1\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-1}a=1\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852e9b8caebcc8c65cdaabf9962747723aa68ad8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.022ex; height:2.676ex;" alt="{\displaystyle a^{-1}a=1\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea08ed932b6883d805e392918b1df37de2a891e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.783ex; height:1.676ex;" alt="{\displaystyle u=}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1u=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mi>u</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1u=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fd1cdf1332d7c71b709b1ab7b136e83d65435ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.945ex; height:2.176ex;" alt="{\displaystyle 1u=}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a^{-1}a)u=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>a</mi> <mo stretchy="false">)</mo> <mi>u</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a^{-1}a)u=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3b371ddd0351a89d88b5c070fc934d48405371c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.385ex; height:3.176ex;" alt="{\displaystyle (a^{-1}a)u=}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{-1}(au)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-1}(au)=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd18bb9cdf5d12ac953398be23b642d1b325b594" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.385ex; height:3.176ex;" alt="{\displaystyle a^{-1}(au)=}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{-1}0=0\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-1}0=0\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc23fe38d6a8cfdd20c3bddd02c3caa0b76cb6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.955ex; height:2.676ex;" alt="{\displaystyle a^{-1}0=0\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{}^{}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{}^{}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12042fe5f808f7aa6db97b69bc365475da25ec39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.237ex; height:2.176ex;" alt="{\displaystyle u_{}^{}=0.}"></span></dd></dl> <p><b>Notación</b> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -au=-(au)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>a</mi> <mi>u</mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -au=-(au)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2983720e321e8080d85f010a1cbae44f38f4c88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.03ex; height:2.843ex;" alt="{\displaystyle -au=-(au)\,}"></span>.</dd></dl> <p><b>Observación</b> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -au=(-a)u=a(-u)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>a</mi> <mi>u</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>u</mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -au=(-a)u=a(-u)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e33f6feb28979e7d75a5351a29bddb491cbb2313" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.306ex; height:2.843ex;" alt="{\displaystyle -au=(-a)u=a(-u)\,}"></span></dd></dl> <dl><dd><ul><li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle au+a(-u)=a(u-u)=a0=0\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>u</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle au+a(-u)=a(u-u)=a0=0\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a20532ecc69b87b6e1ee6f17373485b607c260da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.935ex; height:2.843ex;" alt="{\displaystyle au+a(-u)=a(u-u)=a0=0\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(-u)=-au\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>a</mi> <mi>u</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(-u)=-au\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/481bf20f61a8e01ceb2a8f03b434f568c6fb80d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.03ex; height:2.843ex;" alt="{\displaystyle a(-u)=-au\,}"></span></li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle au+(-a)u=(a-a)u=0u=0\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>u</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>u</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle au+(-a)u=(a-a)u=0u=0\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ca7628c5d0b670b461e79c792eb6d580e6d7a1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.935ex; height:2.843ex;" alt="{\displaystyle au+(-a)u=(a-a)u=0u=0\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-a)u=-au\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>u</mi> <mo>=</mo> <mo>−</mo> <mi>a</mi> <mi>u</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-a)u=-au\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc735c1bbf04043f9dedef26aaf0337e6d5de71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.03ex; height:2.843ex;" alt="{\displaystyle (-a)u=-au\,}"></span></li></ul></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Primer_ejemplo_con_demostración"><span id="Primer_ejemplo_con_demostraci.C3.B3n"></span>Primer ejemplo con demostración</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=6" title="Editar sección: Primer ejemplo con demostración"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se quiere probar que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> es un espacio vectorial sobre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> </p><p>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> juega el papel de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6d3a3cf2cdc8c8b0999c7899352fedb33573cdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.432ex; height:2.176ex;" alt="{\displaystyle V\;}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> el de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1f75b87a7dbaa8542db58d1d8cc0f596676c6dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.711ex; height:2.176ex;" alt="{\displaystyle K\;}"></span>: </p><p>Los elementos: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} \in V=\mathbb {R} ^{2}=\mathbb {R} \times {}\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>∈</mo> <mi>V</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} \in V=\mathbb {R} ^{2}=\mathbb {R} \times {}\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc63a29da258706eceeb797c596d74dca81cd9a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.239ex; height:2.676ex;" alt="{\displaystyle \mathbf {u} \in V=\mathbb {R} ^{2}=\mathbb {R} \times {}\mathbb {R} }"></span></dd></dl> <p>son, de forma genérica: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} =(u_{x},u_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} =(u_{x},u_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c87321f04fc4494a45e7afe5f5b7284c9f9589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.308ex; height:3.009ex;" alt="{\displaystyle \mathbf {u} =(u_{x},u_{y})}"></span></dd></dl> <p>es decir, pares de números reales. Por claridad se conserva la denominación del vector, en este caso <b>u</b>, en sus coordenadas, añadiendo el subíndice <b>x</b> o <b>y</b> para denominar su componente en el eje <b>x</b> o <b>y</b> respectivamente </p><p>En <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6d3a3cf2cdc8c8b0999c7899352fedb33573cdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.432ex; height:2.176ex;" alt="{\displaystyle V\;}"></span> se define la operación suma: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{ccll}+:&{V\times {}V}&\longrightarrow {}&{V}\\&(\mathbf {u} ,\mathbf {v} )&\mapsto &\mathbf {w} =\mathbf {u} +\mathbf {v} \end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>+</mo> <mo>:</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>V</mi> </mrow> </mtd> <mtd> <mo stretchy="false">⟶</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mo stretchy="false">↦</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{ccll}+:&{V\times {}V}&\longrightarrow {}&{V}\\&(\mathbf {u} ,\mathbf {v} )&\mapsto &\mathbf {w} =\mathbf {u} +\mathbf {v} \end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2942bd798c5e2018abda22490eb113b8403ac97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.451ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{ccll}+:&{V\times {}V}&\longrightarrow {}&{V}\\&(\mathbf {u} ,\mathbf {v} )&\mapsto &\mathbf {w} =\mathbf {u} +\mathbf {v} \end{array}}}"></span></dd></dl> <p>donde: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} =(u_{x},u_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} =(u_{x},u_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c87321f04fc4494a45e7afe5f5b7284c9f9589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.308ex; height:3.009ex;" alt="{\displaystyle \mathbf {u} =(u_{x},u_{y})}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =(v_{x},v_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =(v_{x},v_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/646526ba7ad55063e082eb4b1266a4ff309f9f7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.83ex; height:3.009ex;" alt="{\displaystyle \mathbf {v} =(v_{x},v_{y})}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} =(w_{x},w_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} =(w_{x},w_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36aa23621250b0eb383ea285e203b87fdc96bd48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.423ex; height:3.009ex;" alt="{\displaystyle \mathbf {w} =(w_{x},w_{y})}"></span></dd></dl> <p>y la suma de <b>u</b> y <b>v</b> sería: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} +\mathbf {v} =(u_{x},u_{y})+(v_{x},v_{y})=(u_{x}+v_{x},u_{y}+v_{y})=(w_{x},w_{y})=\mathbf {w} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} +\mathbf {v} =(u_{x},u_{y})+(v_{x},v_{y})=(u_{x}+v_{x},u_{y}+v_{y})=(w_{x},w_{y})=\mathbf {w} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e1628c1e0741f8faee130778b2cd08a6012b542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:64.222ex; height:3.009ex;" alt="{\displaystyle \mathbf {u} +\mathbf {v} =(u_{x},u_{y})+(v_{x},v_{y})=(u_{x}+v_{x},u_{y}+v_{y})=(w_{x},w_{y})=\mathbf {w} }"></span></dd></dl> <p>donde: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{l}w_{x}=u_{x}+v_{x}\\w_{y}=u_{y}+v_{y}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{l}w_{x}=u_{x}+v_{x}\\w_{y}=u_{y}+v_{y}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ecbb4dd198b9cb0d6394a367373437627952c56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.329ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{l}w_{x}=u_{x}+v_{x}\\w_{y}=u_{y}+v_{y}\end{array}}}"></span></dd></dl> <p>esto implica que la suma de vectores es interna y bien definida. </p><p>La operación interna suma tiene las propiedades: </p><p>1) La propiedad conmutativa, es decir: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} +\mathbf {v} =\mathbf {v} +\mathbf {u} ,\quad \forall {}\mathbf {u} ,\mathbf {v} \in {}V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} +\mathbf {v} =\mathbf {v} +\mathbf {u} ,\quad \forall {}\mathbf {u} ,\mathbf {v} \in {}V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4011ddd30a150094c575808ef416ae5d75461d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.779ex; height:2.509ex;" alt="{\displaystyle \mathbf {u} +\mathbf {v} =\mathbf {v} +\mathbf {u} ,\quad \forall {}\mathbf {u} ,\mathbf {v} \in {}V}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} +\mathbf {v} =\mathbf {v} +\mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} +\mathbf {v} =\mathbf {v} +\mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17c410bd673049a06868afce5f0167a6d510f3fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.572ex; height:2.176ex;" alt="{\displaystyle \mathbf {u} +\mathbf {v} =\mathbf {v} +\mathbf {u} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u_{x},u_{y})+(v_{x},v_{y})=\mathbf {v} +\mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u_{x},u_{y})+(v_{x},v_{y})=\mathbf {v} +\mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6908b3ff1a82c6841b3d966b4f088efdf586253c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.72ex; height:3.009ex;" alt="{\displaystyle (u_{x},u_{y})+(v_{x},v_{y})=\mathbf {v} +\mathbf {u} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u_{x}+v_{x},u_{y}+v_{y})=\mathbf {v} +\mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u_{x}+v_{x},u_{y}+v_{y})=\mathbf {v} +\mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03fc7fe8e9c09b35a917a78a1f9ef248fe3707b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.717ex; height:3.009ex;" alt="{\displaystyle (u_{x}+v_{x},u_{y}+v_{y})=\mathbf {v} +\mathbf {u} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v_{x}+u_{x},v_{y}+u_{y})=\mathbf {v} +\mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (v_{x}+u_{x},v_{y}+u_{y})=\mathbf {v} +\mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f316f9b5b95ba4f095641b5565a2168a8cf9591e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.717ex; height:3.009ex;" alt="{\displaystyle (v_{x}+u_{x},v_{y}+u_{y})=\mathbf {v} +\mathbf {u} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v_{x},v_{y})+(u_{x},u_{y})=\mathbf {v} +\mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (v_{x},v_{y})+(u_{x},u_{y})=\mathbf {v} +\mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a9d5b2ad353f4c7260b28a78ca554ea3cfa4b66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.72ex; height:3.009ex;" alt="{\displaystyle (v_{x},v_{y})+(u_{x},u_{y})=\mathbf {v} +\mathbf {u} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} +\mathbf {u} =\mathbf {v} +\mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} +\mathbf {u} =\mathbf {v} +\mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56d8b35fa38070f37d682d49a993e50c12628fd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.572ex; height:2.176ex;" alt="{\displaystyle \mathbf {v} +\mathbf {u} =\mathbf {v} +\mathbf {u} }"></span></dd></dl> <p>2) La propiedad asociativa: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {u} +\mathbf {v} )+\mathbf {w} =\mathbf {u} +(\mathbf {v} +\mathbf {w} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {u} +\mathbf {v} )+\mathbf {w} =\mathbf {u} +(\mathbf {v} +\mathbf {w} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/745d560159580bc6338811ed772968955eded61e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.734ex; height:2.843ex;" alt="{\displaystyle (\mathbf {u} +\mathbf {v} )+\mathbf {w} =\mathbf {u} +(\mathbf {v} +\mathbf {w} )}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Big (}(u_{x},u_{y})+(v_{x},v_{y}){\Big )}+(w_{x},w_{y})=(u_{x},u_{y})+{\Big (}(v_{x},v_{y})+(w_{x},w_{y}){\Big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Big (}(u_{x},u_{y})+(v_{x},v_{y}){\Big )}+(w_{x},w_{y})=(u_{x},u_{y})+{\Big (}(v_{x},v_{y})+(w_{x},w_{y}){\Big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b770d3fced409490eb328274f75aa6f1f4ab24b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:66.887ex; height:4.843ex;" alt="{\displaystyle {\Big (}(u_{x},u_{y})+(v_{x},v_{y}){\Big )}+(w_{x},w_{y})=(u_{x},u_{y})+{\Big (}(v_{x},v_{y})+(w_{x},w_{y}){\Big )}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u_{x}+v_{x},u_{y}+v_{y})+(w_{x},w_{y})=(u_{x},u_{y})+(v_{x}+w_{x},v_{y}+w_{y})\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u_{x}+v_{x},u_{y}+v_{y})+(w_{x},w_{y})=(u_{x},u_{y})+(v_{x}+w_{x},v_{y}+w_{y})\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f51dbac176537f32f3069899969bb65e2323361" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:61.975ex; height:3.009ex;" alt="{\displaystyle (u_{x}+v_{x},u_{y}+v_{y})+(w_{x},w_{y})=(u_{x},u_{y})+(v_{x}+w_{x},v_{y}+w_{y})\;}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u_{x}+v_{x}+w_{x},u_{y}+v_{y}+w_{y})=(u_{x}+v_{x}+w_{x},u_{y}+v_{y}+w_{y})\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u_{x}+v_{x}+w_{x},u_{y}+v_{y}+w_{y})=(u_{x}+v_{x}+w_{x},u_{y}+v_{y}+w_{y})\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840c57b2187ce9160276f2c8aee34e1daa6ecee0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:61.97ex; height:3.009ex;" alt="{\displaystyle (u_{x}+v_{x}+w_{x},u_{y}+v_{y}+w_{y})=(u_{x}+v_{x}+w_{x},u_{y}+v_{y}+w_{y})\;}"></span></dd></dl> <p>3) tiene elemento neutro <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} }"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} +\mathbf {0} =\mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} +\mathbf {0} =\mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/815bf8bb614e6436087b608181f264762f1d7636" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.246ex; height:2.343ex;" alt="{\displaystyle \mathbf {u} +\mathbf {0} =\mathbf {u} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u_{x},u_{y})+(0,0)=(u_{x}+0,u_{y}+0)=(u_{x},u_{y})\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u_{x},u_{y})+(0,0)=(u_{x}+0,u_{y}+0)=(u_{x},u_{y})\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adcbaf81218ab997374248a3fab1524a5545e6cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.03ex; height:3.009ex;" alt="{\displaystyle (u_{x},u_{y})+(0,0)=(u_{x}+0,u_{y}+0)=(u_{x},u_{y})\;}"></span></dd></dl> <p>4) tenga elemento opuesto: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} =(u_{x},u_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} =(u_{x},u_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c87321f04fc4494a45e7afe5f5b7284c9f9589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.308ex; height:3.009ex;" alt="{\displaystyle \mathbf {u} =(u_{x},u_{y})}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {-u} =(-u_{x},-u_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold">−</mo> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {-u} =(-u_{x},-u_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3052a379608a57cb12e406f0b40d6e680ab3cd0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.002ex; height:3.009ex;" alt="{\displaystyle \mathbf {-u} =(-u_{x},-u_{y})}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} +(\mathbf {-u} )=(u_{x},u_{y})+(-u_{x},-u_{y})=(u_{x}-u_{x},u_{y}-u_{y})=(0,0)=\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold">−</mo> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} +(\mathbf {-u} )=(u_{x},u_{y})+(-u_{x},-u_{y})=(u_{x}-u_{x},u_{y}-u_{y})=(0,0)=\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/689b75f58d42c9e512bf8dac592c3e7ace5f4a86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:68.788ex; height:3.009ex;" alt="{\displaystyle \mathbf {u} +(\mathbf {-u} )=(u_{x},u_{y})+(-u_{x},-u_{y})=(u_{x}-u_{x},u_{y}-u_{y})=(0,0)=\mathbf {0} }"></span></dd></dl> <p>La operación producto por un escalar: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{ccll}\cdot :&K\times V&\longrightarrow &V\\&({\mathit {a}},\mathbf {u} )&\mapsto &\mathbf {v} ={\mathit {a}}\cdot \mathbf {u} \end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>⋅</mo> <mo>:</mo> </mtd> <mtd> <mi>K</mi> <mo>×</mo> <mi>V</mi> </mtd> <mtd> <mo stretchy="false">⟶</mo> </mtd> <mtd> <mi>V</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mo stretchy="false">↦</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{ccll}\cdot :&K\times V&\longrightarrow &V\\&({\mathit {a}},\mathbf {u} )&\mapsto &\mathbf {v} ={\mathit {a}}\cdot \mathbf {u} \end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf8c64d560bbbae78f917ffd6e577a6d8f4ef080" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.019ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{ccll}\cdot :&K\times V&\longrightarrow &V\\&({\mathit {a}},\mathbf {u} )&\mapsto &\mathbf {v} ={\mathit {a}}\cdot \mathbf {u} \end{array}}}"></span></dd></dl> <p>El producto de <b>a</b> y <b>u</b> será: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot \mathbf {u} =a\cdot (u_{x},u_{y})=(a\cdot u_{x},a\cdot u_{y})=(v_{x},v_{y})=\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mi>a</mi> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot \mathbf {u} =a\cdot (u_{x},u_{y})=(a\cdot u_{x},a\cdot u_{y})=(v_{x},v_{y})=\mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b72fda74413fe71cbe1b10d647ed52bad934728" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:49.653ex; height:3.009ex;" alt="{\displaystyle {\mathit {a}}\cdot \mathbf {u} =a\cdot (u_{x},u_{y})=(a\cdot u_{x},a\cdot u_{y})=(v_{x},v_{y})=\mathbf {v} }"></span></dd></dl> <p>donde: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{l}v_{x}=a\cdot u_{x}\\v_{y}=a\cdot u_{y}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{l}v_{x}=a\cdot u_{x}\\v_{y}=a\cdot u_{y}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce550fde9ea064d489a75ec5df28cec0e2414d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.561ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{l}v_{x}=a\cdot u_{x}\\v_{y}=a\cdot u_{y}\end{array}}}"></span></dd></dl> <p>esto implica que la multiplicación de vector por escalar es externa y aun así está bien definida. </p><p>5) tenga la propiedad asociativa: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot ({\mathit {b}}\cdot \mathbf {u} )=({\mathit {a}}\cdot {\mathit {b}})\cdot \mathbf {u} ,\quad \forall {}{\mathit {a}},{\mathit {b}}\in {}K,\quad \forall {}\mathbf {u} \in {}V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>K</mi> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot ({\mathit {b}}\cdot \mathbf {u} )=({\mathit {a}}\cdot {\mathit {b}})\cdot \mathbf {u} ,\quad \forall {}{\mathit {a}},{\mathit {b}}\in {}K,\quad \forall {}\mathbf {u} \in {}V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f6b19aff74b36bd023676c081511f7f29bb746d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.529ex; height:2.843ex;" alt="{\displaystyle {\mathit {a}}\cdot ({\mathit {b}}\cdot \mathbf {u} )=({\mathit {a}}\cdot {\mathit {b}})\cdot \mathbf {u} ,\quad \forall {}{\mathit {a}},{\mathit {b}}\in {}K,\quad \forall {}\mathbf {u} \in {}V}"></span></dd></dl> <p>Esto es: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot ({\mathit {b}}\cdot \mathbf {u} )=({\mathit {a}}\cdot {\mathit {b}})\cdot \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot ({\mathit {b}}\cdot \mathbf {u} )=({\mathit {a}}\cdot {\mathit {b}})\cdot \mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1c6dfe098b75ce87d7faa1ef118eab65741eea7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.919ex; height:2.843ex;" alt="{\displaystyle {\mathit {a}}\cdot ({\mathit {b}}\cdot \mathbf {u} )=({\mathit {a}}\cdot {\mathit {b}})\cdot \mathbf {u} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot ({\mathit {b}}\cdot (u_{x},u_{y}))=({\mathit {a}}\cdot {\mathit {b}})\cdot (u_{x},u_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot ({\mathit {b}}\cdot (u_{x},u_{y}))=({\mathit {a}}\cdot {\mathit {b}})\cdot (u_{x},u_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/011e32f50f2c02307d0eea979ef09ca464e66ad3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.397ex; height:3.009ex;" alt="{\displaystyle {\mathit {a}}\cdot ({\mathit {b}}\cdot (u_{x},u_{y}))=({\mathit {a}}\cdot {\mathit {b}})\cdot (u_{x},u_{y})}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot ({\mathit {b}}\cdot u_{x},{\mathit {b}}\cdot u_{y})=({\mathit {a}}\cdot {\mathit {b}})\cdot (u_{x},u_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot ({\mathit {b}}\cdot u_{x},{\mathit {b}}\cdot u_{y})=({\mathit {a}}\cdot {\mathit {b}})\cdot (u_{x},u_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96c31ae84e3435101d2e09853e26094d2a5a2234" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.337ex; height:3.009ex;" alt="{\displaystyle {\mathit {a}}\cdot ({\mathit {b}}\cdot u_{x},{\mathit {b}}\cdot u_{y})=({\mathit {a}}\cdot {\mathit {b}})\cdot (u_{x},u_{y})}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathit {a}}\cdot {\mathit {b}}\cdot u_{x},{\mathit {a}}\cdot {\mathit {b}}\cdot u_{y})=({\mathit {a}}\cdot {\mathit {b}}\cdot u_{x},{\mathit {a}}\cdot {\mathit {b}}\cdot u_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathit {a}}\cdot {\mathit {b}}\cdot u_{x},{\mathit {a}}\cdot {\mathit {b}}\cdot u_{y})=({\mathit {a}}\cdot {\mathit {b}}\cdot u_{x},{\mathit {a}}\cdot {\mathit {b}}\cdot u_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/408b0185e0d23c8e0ca99001c6571092edf8a199" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.01ex; height:3.009ex;" alt="{\displaystyle ({\mathit {a}}\cdot {\mathit {b}}\cdot u_{x},{\mathit {a}}\cdot {\mathit {b}}\cdot u_{y})=({\mathit {a}}\cdot {\mathit {b}}\cdot u_{x},{\mathit {a}}\cdot {\mathit {b}}\cdot u_{y})}"></span></dd></dl> <p>6) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {1}}\in {}R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {1}}\in {}R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19d69264ee494dd73e9a8ef1c54c9a489004cf6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.793ex; height:2.176ex;" alt="{\displaystyle {\mathit {1}}\in {}R}"></span> sea elemento neutro en el producto: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {1}}\cdot \mathbf {u} =\mathbf {u} ,\quad \forall {}\mathbf {u} \in {}V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {1}}\cdot \mathbf {u} =\mathbf {u} ,\quad \forall {}\mathbf {u} \in {}V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e5119c38e10e4a1453187ea4f04305904a6e42e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.698ex; height:2.509ex;" alt="{\displaystyle {\mathit {1}}\cdot \mathbf {u} =\mathbf {u} ,\quad \forall {}\mathbf {u} \in {}V}"></span></dd></dl> <p>Que resulta: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {1}}\cdot \mathbf {u} =\mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {1}}\cdot \mathbf {u} =\mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/644519875c4b95879ab4d7ab5d06600a059843fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.936ex; height:2.176ex;" alt="{\displaystyle {\mathit {1}}\cdot \mathbf {u} =\mathbf {u} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {1}}\cdot (u_{x},u_{y})=\mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {1}}\cdot (u_{x},u_{y})=\mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d05f6f374e78915d1f01bcd30cb2d9fc7e078ed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.175ex; height:3.009ex;" alt="{\displaystyle {\mathit {1}}\cdot (u_{x},u_{y})=\mathbf {u} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathit {1}}\cdot u_{x},{\mathit {1}}\cdot u_{y})=\mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-mathit" mathvariant="italic">1</mn> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathit {1}}\cdot u_{x},{\mathit {1}}\cdot u_{y})=\mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/760ef9dcb5c6eeadfc3aa68cbf58e66aed71bd7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.042ex; height:3.009ex;" alt="{\displaystyle ({\mathit {1}}\cdot u_{x},{\mathit {1}}\cdot u_{y})=\mathbf {u} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u_{x},u_{y})=\mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u_{x},u_{y})=\mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d8ec746461e483d4b3c3916fca979fbce078bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.308ex; height:3.009ex;" alt="{\displaystyle (u_{x},u_{y})=\mathbf {u} }"></span></dd></dl> <p>Que tiene la propiedad distributiva: </p><p>7) distributiva por la izquierda: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot (\mathbf {u} +\mathbf {v} )={\mathit {a}}\cdot \mathbf {u} +{\mathit {a}}\cdot \mathbf {v} ,\quad \forall {}{\mathit {a}}\in {}R,\quad \forall {}\mathbf {u} ,\mathbf {v} \in {}V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>R</mi> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot (\mathbf {u} +\mathbf {v} )={\mathit {a}}\cdot \mathbf {u} +{\mathit {a}}\cdot \mathbf {v} ,\quad \forall {}{\mathit {a}}\in {}R,\quad \forall {}\mathbf {u} ,\mathbf {v} \in {}V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e8bdcba61e8540824ae9fe8863bc30601b7aa98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.631ex; height:2.843ex;" alt="{\displaystyle {\mathit {a}}\cdot (\mathbf {u} +\mathbf {v} )={\mathit {a}}\cdot \mathbf {u} +{\mathit {a}}\cdot \mathbf {v} ,\quad \forall {}{\mathit {a}}\in {}R,\quad \forall {}\mathbf {u} ,\mathbf {v} \in {}V}"></span></dd></dl> <p>En este caso tenemos: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot (\mathbf {u} +\mathbf {v} )={\mathit {a}}\cdot \mathbf {u} +{\mathit {a}}\cdot \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot (\mathbf {u} +\mathbf {v} )={\mathit {a}}\cdot \mathbf {u} +{\mathit {a}}\cdot \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/788acc9745ec9e85bebf7a28b63b3652749945c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.982ex; height:2.843ex;" alt="{\displaystyle {\mathit {a}}\cdot (\mathbf {u} +\mathbf {v} )={\mathit {a}}\cdot \mathbf {u} +{\mathit {a}}\cdot \mathbf {v} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot ((u_{x},u_{y})+(v_{x},v_{y}))={\mathit {a}}\cdot (u_{x},u_{y})+{\mathit {a}}\cdot (v_{x},v_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot ((u_{x},u_{y})+(v_{x},v_{y}))={\mathit {a}}\cdot (u_{x},u_{y})+{\mathit {a}}\cdot (v_{x},v_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/963efadcf368e927d75a822bd29d4ee2406f0384" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:49.279ex; height:3.009ex;" alt="{\displaystyle {\mathit {a}}\cdot ((u_{x},u_{y})+(v_{x},v_{y}))={\mathit {a}}\cdot (u_{x},u_{y})+{\mathit {a}}\cdot (v_{x},v_{y})}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot (u_{x}+v_{x},u_{y}+v_{y})=({\mathit {a}}\cdot u_{x},{\mathit {a}}\cdot u_{y})+({\mathit {a}}\cdot v_{x},{\mathit {a}}\cdot v_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot (u_{x}+v_{x},u_{y}+v_{y})=({\mathit {a}}\cdot u_{x},{\mathit {a}}\cdot u_{y})+({\mathit {a}}\cdot v_{x},{\mathit {a}}\cdot v_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3e624a85d824979455976aebc3d84733897b7d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:53.201ex; height:3.009ex;" alt="{\displaystyle {\mathit {a}}\cdot (u_{x}+v_{x},u_{y}+v_{y})=({\mathit {a}}\cdot u_{x},{\mathit {a}}\cdot u_{y})+({\mathit {a}}\cdot v_{x},{\mathit {a}}\cdot v_{y})}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {a}}\cdot (u_{x}+v_{x},u_{y}+v_{y})=({\mathit {a}}\cdot u_{x}+{\mathit {a}}\cdot v_{x},{\mathit {a}}\cdot u_{y}+{\mathit {a}}\cdot v_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathit {a}}\cdot (u_{x}+v_{x},u_{y}+v_{y})=({\mathit {a}}\cdot u_{x}+{\mathit {a}}\cdot v_{x},{\mathit {a}}\cdot u_{y}+{\mathit {a}}\cdot v_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0513b99c90d30d79d8c22461983b1cc941f9cff4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:53.198ex; height:3.009ex;" alt="{\displaystyle {\mathit {a}}\cdot (u_{x}+v_{x},u_{y}+v_{y})=({\mathit {a}}\cdot u_{x}+{\mathit {a}}\cdot v_{x},{\mathit {a}}\cdot u_{y}+{\mathit {a}}\cdot v_{y})}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathit {a}}\cdot (u_{x}+v_{x}),{\mathit {a}}\cdot (u_{y}+v_{y}))=({\mathit {a}}\cdot (u_{x}+v_{x}),{\mathit {a}}\cdot (u_{y}+v_{y}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathit {a}}\cdot (u_{x}+v_{x}),{\mathit {a}}\cdot (u_{y}+v_{y}))=({\mathit {a}}\cdot (u_{x}+v_{x}),{\mathit {a}}\cdot (u_{y}+v_{y}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab73d8c6805509621a9eef6c9291e302e32e0825" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:57.569ex; height:3.009ex;" alt="{\displaystyle ({\mathit {a}}\cdot (u_{x}+v_{x}),{\mathit {a}}\cdot (u_{y}+v_{y}))=({\mathit {a}}\cdot (u_{x}+v_{x}),{\mathit {a}}\cdot (u_{y}+v_{y}))}"></span></dd></dl> <p>8) distributiva por la derecha: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot \mathbf {u} ={\mathit {a}}\cdot \mathbf {u} +{\mathit {b}}\cdot \mathbf {u} ,\quad \forall {}{\mathit {a}},{\mathit {b}}\in {}R,\quad \forall {}\mathbf {u} \in {}V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>R</mi> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot \mathbf {u} ={\mathit {a}}\cdot \mathbf {u} +{\mathit {b}}\cdot \mathbf {u} ,\quad \forall {}{\mathit {a}},{\mathit {b}}\in {}R,\quad \forall {}\mathbf {u} \in {}V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31086b0de59c76832110cf2696c293914f8f374b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.904ex; height:2.843ex;" alt="{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot \mathbf {u} ={\mathit {a}}\cdot \mathbf {u} +{\mathit {b}}\cdot \mathbf {u} ,\quad \forall {}{\mathit {a}},{\mathit {b}}\in {}R,\quad \forall {}\mathbf {u} \in {}V}"></span></dd></dl> <p>Que en este caso tenemos: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot \mathbf {u} ={\mathit {a}}\cdot \mathbf {u} +{\mathit {b}}\cdot \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot \mathbf {u} ={\mathit {a}}\cdot \mathbf {u} +{\mathit {b}}\cdot \mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/373557631f9f1ec13ac9936cdc4911eb0a6bd93d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.597ex; height:2.843ex;" alt="{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot \mathbf {u} ={\mathit {a}}\cdot \mathbf {u} +{\mathit {b}}\cdot \mathbf {u} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot (u_{x},u_{y})={\mathit {a}}\cdot (u_{x},u_{y})+{\mathit {b}}\cdot (u_{x},u_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot (u_{x},u_{y})={\mathit {a}}\cdot (u_{x},u_{y})+{\mathit {b}}\cdot (u_{x},u_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f9f4e93b23fe277a5493b2aa3913232ecb555d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.314ex; height:3.009ex;" alt="{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot (u_{x},u_{y})={\mathit {a}}\cdot (u_{x},u_{y})+{\mathit {b}}\cdot (u_{x},u_{y})}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot (u_{x},u_{y})=({\mathit {a}}\cdot u_{x},{\mathit {a}}\cdot u_{y})+({\mathit {b}}\cdot u_{x},{\mathit {b}}\cdot u_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot (u_{x},u_{y})=({\mathit {a}}\cdot u_{x},{\mathit {a}}\cdot u_{y})+({\mathit {b}}\cdot u_{x},{\mathit {b}}\cdot u_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d594458d5174de4bd5df53cbde155472d44f11d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.93ex; height:3.009ex;" alt="{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot (u_{x},u_{y})=({\mathit {a}}\cdot u_{x},{\mathit {a}}\cdot u_{y})+({\mathit {b}}\cdot u_{x},{\mathit {b}}\cdot u_{y})}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot (u_{x},u_{y})=({\mathit {a}}\cdot u_{x}+{\mathit {b}}\cdot u_{x},{\mathit {a}}\cdot u_{y}+{\mathit {b}}\cdot u_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot (u_{x},u_{y})=({\mathit {a}}\cdot u_{x}+{\mathit {b}}\cdot u_{x},{\mathit {a}}\cdot u_{y}+{\mathit {b}}\cdot u_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b88350dafbcda229ae53c39e615d4e2143b5e1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.927ex; height:3.009ex;" alt="{\displaystyle ({\mathit {a}}+{\mathit {b}})\cdot (u_{x},u_{y})=({\mathit {a}}\cdot u_{x}+{\mathit {b}}\cdot u_{x},{\mathit {a}}\cdot u_{y}+{\mathit {b}}\cdot u_{y})}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (({\mathit {a}}+{\mathit {b}})\cdot u_{x},({\mathit {a}}+{\mathit {b}})\cdot u_{y})=(({\mathit {a}}+{\mathit {b}})\cdot u_{x},({\mathit {a}}+{\mathit {b}})\cdot u_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">a</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (({\mathit {a}}+{\mathit {b}})\cdot u_{x},({\mathit {a}}+{\mathit {b}})\cdot u_{y})=(({\mathit {a}}+{\mathit {b}})\cdot u_{x},({\mathit {a}}+{\mathit {b}})\cdot u_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa3d13019f4838e901ee42f97872f0cd346e30cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:52.893ex; height:3.009ex;" alt="{\displaystyle (({\mathit {a}}+{\mathit {b}})\cdot u_{x},({\mathit {a}}+{\mathit {b}})\cdot u_{y})=(({\mathit {a}}+{\mathit {b}})\cdot u_{x},({\mathit {a}}+{\mathit {b}})\cdot u_{y})}"></span></dd></dl> <p>Queda demostrado que es espacio vectorial. </p> <div class="mw-heading mw-heading2"><h2 id="Ejemplos">Ejemplos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=7" title="Editar sección: Ejemplos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Los_cuerpos">Los cuerpos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=8" title="Editar sección: Los cuerpos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Todo cuerpo es un espacio vectorial sobre él mismo, usando como producto por escalar el producto del cuerpo. </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> es un espacio vectorial de dimensión uno sobre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>.</li></ul> <p>Todo cuerpo es un espacio vectorial sobre su <a href="/wiki/Cuerpo_(matem%C3%A1tica)" class="mw-redirect" title="Cuerpo (matemática)">subcuerpo</a>, usando como producto por escalar el producto del cuerpo. </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> es un espacio vectorial sobre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> es un espacio vectorial sobre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Sucesiones_sobre_un_cuerpo_K">Sucesiones sobre un cuerpo K</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=9" title="Editar sección: Sucesiones sobre un cuerpo K"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El <a href="/wiki/Vector_(espacio_eucl%C3%ADdeo)" class="mw-redirect" title="Vector (espacio euclídeo)">espacio vectorial más conocido</a> notado como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{}^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e52d58b4936de3abbe7a6ecf90a7097f1d33536a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.312ex; height:2.343ex;" alt="{\displaystyle K_{}^{n}}"></span>, donde <i>n</i>>0 es un <a href="/wiki/Entero" class="mw-redirect" title="Entero">entero</a>, tiene como elementos <a href="/wiki/Tupla" title="Tupla"><i>n</i>-tuplas</a>, es decir, <a href="/wiki/Sucesi%C3%B3n_matem%C3%A1tica" class="mw-redirect" title="Sucesión matemática">sucesiones</a> finitas de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a6c0f78592eaf8c5b477d9551ea3adffeea33fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K_{}^{}}"></span> de longitud <i>n</i> con las operaciones: </p> <dl><dd>(<i>u</i><sub>1</sub>, <i>u</i><sub>2</sub>, ..., <i>u</i><sub><i>n</i></sub>)+(<i>v</i><sub>1</sub>, <i>v</i><sub>2</sub>, ..., <i>v</i><sub><i>n</i></sub>)=(<i>u</i><sub>1</sub>+<i>v</i><sub>1</sub>, <i>u</i><sub>2</sub>+<i>v</i><sub>2</sub>, ..., <i>u</i><sub><i>n</i></sub>+<i>v</i><sub>n</sub>).</dd></dl> <dl><dd>a(<i>u</i><sub>1</sub>, <i>u</i><sub>2</sub>, ..., <i>u</i><sub><i>n</i></sub>)=(<i>au</i><sub>1</sub>, <i>au</i><sub>2</sub>, ..., <i>au</i><sub><i>n</i></sub>).</dd></dl> <p>Las sucesiones infinitas de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bfd902d22190f21473bfeb6cbfe09bade30e4f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K^{}}"></span> son espacios vectoriales con las operaciones: </p> <dl><dd>(<i>u</i><sub>1</sub>, <i>u</i><sub>2</sub>, ..., <i>u</i><sub><i>n</i></sub>, ...)+(<i>v</i><sub>1</sub>, <i>v</i><sub>2</sub>, ..., <i>v</i><sub><i>n</i></sub>, ...)=(<i>u</i><sub>1</sub>+<i>v</i><sub>1</sub>, <i>u</i><sub>2</sub>+<i>v</i><sub>2</sub>, ..., <i>u</i><sub><i>n</i></sub>+<i>v</i><sub>n</sub>, ...).</dd></dl> <dl><dd>a(<i>u</i><sub>1</sub>, <i>u</i><sub>2</sub>, ..., <i>u</i><sub><i>n</i></sub>, ...)=(<i>au</i><sub>1</sub>, <i>au</i><sub>2</sub>, ..., <i>au</i><sub><i>n</i></sub>, ...).</dd></dl> <p>El espacio de las <a href="/wiki/Matriz_(matem%C3%A1ticas)" class="mw-redirect" title="Matriz (matemáticas)">matrices</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d82325a2a02ad79bc7c347ba9702ad46eb0de824" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle n\times m}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{n\times m}(K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>×</mo> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{n\times m}(K)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f67a147c95291bbc5702a9e8cd0ea16ab9489fb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.069ex; height:2.843ex;" alt="{\displaystyle M_{n\times m}(K)}"></span>, sobre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bfd902d22190f21473bfeb6cbfe09bade30e4f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K^{}}"></span>, con las operaciones: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}x_{1,1}&\cdots &x_{1,m}\\\vdots &&\vdots \\x_{n,1}&\cdots &x_{n,m}\end{pmatrix}}+{\begin{pmatrix}y_{1,1}&\cdots &y_{1,m}\\\vdots &&\vdots \\y_{n,1}&\cdots &y_{n,m}\end{pmatrix}}={\begin{pmatrix}x_{1,1}+y_{1,1}&\cdots &x_{1,m}+y_{1,m}\\\vdots &&\vdots \\x_{n,1}+y_{n,1}&\cdots &x_{n,m}+y_{n,m}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>m</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>m</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>m</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}x_{1,1}&\cdots &x_{1,m}\\\vdots &&\vdots \\x_{n,1}&\cdots &x_{n,m}\end{pmatrix}}+{\begin{pmatrix}y_{1,1}&\cdots &y_{1,m}\\\vdots &&\vdots \\y_{n,1}&\cdots &y_{n,m}\end{pmatrix}}={\begin{pmatrix}x_{1,1}+y_{1,1}&\cdots &x_{1,m}+y_{1,m}\\\vdots &&\vdots \\x_{n,1}+y_{n,1}&\cdots &x_{n,m}+y_{n,m}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/650e873e946b21cc0f9a0896bf9a968761aa018b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.62ex; margin-bottom: -0.218ex; width:80.521ex; height:10.843ex;" alt="{\displaystyle {\begin{pmatrix}x_{1,1}&\cdots &x_{1,m}\\\vdots &&\vdots \\x_{n,1}&\cdots &x_{n,m}\end{pmatrix}}+{\begin{pmatrix}y_{1,1}&\cdots &y_{1,m}\\\vdots &&\vdots \\y_{n,1}&\cdots &y_{n,m}\end{pmatrix}}={\begin{pmatrix}x_{1,1}+y_{1,1}&\cdots &x_{1,m}+y_{1,m}\\\vdots &&\vdots \\x_{n,1}+y_{n,1}&\cdots &x_{n,m}+y_{n,m}\end{pmatrix}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\begin{pmatrix}x_{1,1}&\cdots &x_{1,m}\\\vdots &&\vdots \\x_{n,1}&\cdots &x_{n,m}\end{pmatrix}}={\begin{pmatrix}ax_{1,1}&\cdots &ax_{1,m}\\\vdots &&\vdots \\ax_{n,1}&\cdots &ax_{n,m}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>m</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>a</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>m</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>a</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\begin{pmatrix}x_{1,1}&\cdots &x_{1,m}\\\vdots &&\vdots \\x_{n,1}&\cdots &x_{n,m}\end{pmatrix}}={\begin{pmatrix}ax_{1,1}&\cdots &ax_{1,m}\\\vdots &&\vdots \\ax_{n,1}&\cdots &ax_{n,m}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98624cd28e3d9998bd04f2963ba8b3354ec4d8d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.62ex; margin-bottom: -0.218ex; width:47.713ex; height:10.843ex;" alt="{\displaystyle a{\begin{pmatrix}x_{1,1}&\cdots &x_{1,m}\\\vdots &&\vdots \\x_{n,1}&\cdots &x_{n,m}\end{pmatrix}}={\begin{pmatrix}ax_{1,1}&\cdots &ax_{1,m}\\\vdots &&\vdots \\ax_{n,1}&\cdots &ax_{n,m}\end{pmatrix}}}"></span></dd></dl> <p>También son espacios vectoriales cualquier agrupación de elementos de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a6c0f78592eaf8c5b477d9551ea3adffeea33fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K_{}^{}}"></span> en las cuales se defina las operaciones suma y producto entre estas agrupaciones, elemento a elemento, similar al de matrices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d82325a2a02ad79bc7c347ba9702ad46eb0de824" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle n\times m}"></span>, así por ejemplo tenemos las cajas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times m\times r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>m</mi> <mo>×</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times m\times r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/942173f290731b6865c3f5098bf318e3e41b5c40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.165ex; height:1.676ex;" alt="{\displaystyle n\times m\times r}"></span> sobre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a6c0f78592eaf8c5b477d9551ea3adffeea33fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K_{}^{}}"></span> que aparecen en el desarrollo de Taylor de orden 3 de una función genérica. </p> <div class="mw-heading mw-heading3"><h3 id="Espacios_de_aplicaciones_sobre_un_cuerpo">Espacios de aplicaciones sobre un cuerpo</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=10" title="Editar sección: Espacios de aplicaciones sobre un cuerpo"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El conjunto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba69787528be36f0312673a5ffe86596b55f4a24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F_{}^{}}"></span> de las aplicaciones <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:M\rightarrow K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:M\rightarrow K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d705d1eace610a9ef03323a5b4439313d1586a92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.338ex; height:2.509ex;" alt="{\displaystyle f:M\rightarrow K}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bfd902d22190f21473bfeb6cbfe09bade30e4f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K^{}}"></span> un cuerpo y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/429ab5691226ef277cab862679ad128d2fd8885d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M_{}^{}}"></span> un conjunto, también forman espacios vectoriales mediante la suma y la multiplicación habitual: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall f,g\in F,\;\forall a\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <mi>F</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall f,g\in F,\;\forall a\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06a6e087ff9f1f00c5646564c40e40b27204f670" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.411ex; height:2.509ex;" alt="{\displaystyle \forall f,g\in F,\;\forall a\in K}"></span></dd></dl> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}(f+g)(w)&:=f(w)+g(w)_{}^{},\\\;\;\;\;(af)(w)&:=a(f)(w)_{}^{}.\;\;\;\;\;\;\;\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>:=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>w</mi> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>a</mi> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>:=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>w</mi> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo>.</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}(f+g)(w)&:=f(w)+g(w)_{}^{},\\\;\;\;\;(af)(w)&:=a(f)(w)_{}^{}.\;\;\;\;\;\;\;\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f2f410646869e67544ef7a0e50423d7af298849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.646ex; height:6.176ex;" alt="{\displaystyle {\begin{matrix}(f+g)(w)&:=f(w)+g(w)_{}^{},\\\;\;\;\;(af)(w)&:=a(f)(w)_{}^{}.\;\;\;\;\;\;\;\end{matrix}}}"></span></center> <div class="mw-heading mw-heading4"><h4 id="Los_polinomios">Los polinomios</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=11" title="Editar sección: Los polinomios"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Archivo:FuncionesComoEV.GIF" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/FuncionesComoEV.GIF/200px-FuncionesComoEV.GIF" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/FuncionesComoEV.GIF/300px-FuncionesComoEV.GIF 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/FuncionesComoEV.GIF/400px-FuncionesComoEV.GIF 2x" data-file-width="431" data-file-height="432" /></a><figcaption>Suma de <i>f(x)=x+x<sup>2</sup></i> y <i>g(x)=-x<sup>2</sup></i>.</figcaption></figure> <p>El <a href="/wiki/Anillo_de_polinomios" title="Anillo de polinomios">espacio vectorial</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[x]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a9e6c2ac2830d6a9abe078b47450777c41d69a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.689ex; height:2.843ex;" alt="{\displaystyle K[x]}"></span> formado por <a href="/wiki/Funciones_polin%C3%B3micas" class="mw-redirect" title="Funciones polinómicas">funciones polinómicas</a>, veámoslo: </p> <dl><dd>Expresión general: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)=r_{n}x^{n}+r_{n-1}x_{}^{n-1}+...+r_{1}x+r_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)=r_{n}x^{n}+r_{n-1}x_{}^{n-1}+...+r_{1}x+r_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469c157abbb75f739ff3373bb84623762076f6df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:39.262ex; height:3.176ex;" alt="{\displaystyle p(x)=r_{n}x^{n}+r_{n-1}x_{}^{n-1}+...+r_{1}x+r_{0}}"></span>,donde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n},\;...,r_{0}\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n},\;...,r_{0}\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50f44898f833e5203be0ceb99ab35820662fb956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.091ex; height:2.509ex;" alt="{\displaystyle r_{n},\;...,r_{0}\in K}"></span>, considérese <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall i>n\;r_{i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>i</mi> <mo>></mo> <mi>n</mi> <mspace width="thickmathspace" /> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall i>n\;r_{i}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4b58bde634187f988fe6e7b0f7edce560e43ed2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.343ex; height:2.509ex;" alt="{\displaystyle \forall i>n\;r_{i}=0}"></span>.</dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)+q(x)=(r_{n}x^{n}+r_{n-1}x^{n-1}+...+r_{1}x+r_{0}^{})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)+q(x)=(r_{n}x^{n}+r_{n-1}x^{n-1}+...+r_{1}x+r_{0}^{})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ce33d4d4f03c9b013555dfc8871fba3c573e233" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:48.12ex; height:3.176ex;" alt="{\displaystyle p(x)+q(x)=(r_{n}x^{n}+r_{n-1}x^{n-1}+...+r_{1}x+r_{0}^{})}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +(s_{m}x^{m}+s_{m-1}x^{m-1}+...+s_{1}x+s_{0}^{})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +(s_{m}x^{m}+s_{m-1}x^{m-1}+...+s_{1}x+s_{0}^{})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f123c65d2fdedf2d6830271cdf9531298bca322f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.377ex; height:3.176ex;" alt="{\displaystyle +(s_{m}x^{m}+s_{m-1}x^{m-1}+...+s_{1}x+s_{0}^{})}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =..._{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mo>.</mo> <mo>.</mo> <msubsup> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =..._{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ecd8ba588ac8febca13d7fc90597c1a92155c3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.523ex; height:1.509ex;" alt="{\displaystyle =..._{}^{}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =(t_{M}x^{M}+t_{M-1}x^{M-1}+...+t_{1}x+t_{0}^{})=(p+q)(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =(t_{M}x^{M}+t_{M-1}x^{M-1}+...+t_{1}x+t_{0}^{})=(p+q)(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d57e26defab0dca20b9ace79caa140ab047be7bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.281ex; height:3.176ex;" alt="{\displaystyle =(t_{M}x^{M}+t_{M-1}x^{M-1}+...+t_{1}x+t_{0}^{})=(p+q)(x)}"></span>, donde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=\max\{m,\;n\}_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mi>m</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>n</mi> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\max\{m,\;n\}_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/418312556a724726eeaee7e4ed7da9af4236559b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.306ex; height:2.843ex;" alt="{\displaystyle M=\max\{m,\;n\}_{}^{}}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}=r_{i}+s_{i}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}=r_{i}+s_{i}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b806ad70a710d77a6703c29f7da135c7d6a7c5af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.317ex; height:2.343ex;" alt="{\displaystyle t_{i}=r_{i}+s_{i}^{}}"></span>,</dd></dl></dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(p(x))=a(r_{n}x^{n}+r_{n-1}x^{n-1}+...+r_{1}x+r_{0}^{})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(p(x))=a(r_{n}x^{n}+r_{n-1}x^{n-1}+...+r_{1}x+r_{0}^{})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d61a3d20f61ec5dba56071aff3813d96e05fe003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.251ex; height:3.176ex;" alt="{\displaystyle a(p(x))=a(r_{n}x^{n}+r_{n-1}x^{n-1}+...+r_{1}x+r_{0}^{})}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =(ar_{n}x^{n}+ar_{n-1}x^{n-1}+...+ar_{1}x+ar_{0}^{})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mi>a</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <mi>a</mi> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =(ar_{n}x^{n}+ar_{n-1}x^{n-1}+...+ar_{1}x+ar_{0}^{})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093300a3acf73468115dcf5bfc76dd126382f2c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.948ex; height:3.176ex;" alt="{\displaystyle =(ar_{n}x^{n}+ar_{n-1}x^{n-1}+...+ar_{1}x+ar_{0}^{})}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =t_{n}x^{n}+t_{n-1}x^{n-1}+...+t_{1}x+t_{0}^{}=(ap)(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =t_{n}x^{n}+t_{n-1}x^{n-1}+...+t_{1}x+t_{0}^{}=(ap)(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50e28496583120afb386da1e9d14abe0c0762819" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.829ex; height:3.176ex;" alt="{\displaystyle =t_{n}x^{n}+t_{n-1}x^{n-1}+...+t_{1}x+t_{0}^{}=(ap)(x)}"></span>.</dd></dl></dd></dl> <p>Las <a href="/wiki/Series_de_potencias" class="mw-redirect" title="Series de potencias">series de potencias</a> son similares, salvo que se permiten infinitos términos distintos de cero. </p> <div class="mw-heading mw-heading4"><h4 id="Funciones_trigonométricas"><span id="Funciones_trigonom.C3.A9tricas"></span>Funciones trigonométricas</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=12" title="Editar sección: Funciones trigonométricas"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Las funciones trigonométricas forman espacios vectoriales con las siguientes operaciones: </p> <dl><dd>Expresión general: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=a_{f}^{}\sum _{i=1}^{n}(b_{f,i}{\mbox{sen}}(ix)+c_{f,i}\cos(ix))\in L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>sen</mtext> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=a_{f}^{}\sum _{i=1}^{n}(b_{f,i}{\mbox{sen}}(ix)+c_{f,i}\cos(ix))\in L^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/513c76effb977131980b3c30f3329e4fd102023f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.701ex; height:6.843ex;" alt="{\displaystyle f(x)=a_{f}^{}\sum _{i=1}^{n}(b_{f,i}{\mbox{sen}}(ix)+c_{f,i}\cos(ix))\in L^{2}}"></span></dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f+g)(x):=f(x)+g(x)_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f+g)(x):=f(x)+g(x)_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0af6235823fe93f855cc7d0522546bcd4bb4184" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.442ex; height:2.843ex;" alt="{\displaystyle (f+g)(x):=f(x)+g(x)_{}^{}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =a_{f}\sum _{i=1}^{n}(b_{f,i}{\mbox{sen}}(ix)+c_{f,i}\cos(ix))+a_{g}\sum _{i=1}^{n}(b_{g,i}{\mbox{sen}}(ix)+c_{g,i}\cos(ix))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>sen</mtext> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>sen</mtext> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =a_{f}\sum _{i=1}^{n}(b_{f,i}{\mbox{sen}}(ix)+c_{f,i}\cos(ix))+a_{g}\sum _{i=1}^{n}(b_{g,i}{\mbox{sen}}(ix)+c_{g,i}\cos(ix))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/611cc11cd02f7e8ea8c2fa19029862833e00f2a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:68.363ex; height:6.843ex;" alt="{\displaystyle =a_{f}\sum _{i=1}^{n}(b_{f,i}{\mbox{sen}}(ix)+c_{f,i}\cos(ix))+a_{g}\sum _{i=1}^{n}(b_{g,i}{\mbox{sen}}(ix)+c_{g,i}\cos(ix))}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =(a_{f}+a_{g})\sum _{i=1}^{n}((b_{f,i}+b_{g,i}){\mbox{sen}}(ix)+(c_{f,i}+c_{g,i})\cos(ix))\in L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>sen</mtext> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =(a_{f}+a_{g})\sum _{i=1}^{n}((b_{f,i}+b_{g,i}){\mbox{sen}}(ix)+(c_{f,i}+c_{g,i})\cos(ix))\in L^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a68b37066cd229ca0fe1668c0b6f804c3cbf143b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:61.935ex; height:6.843ex;" alt="{\displaystyle =(a_{f}+a_{g})\sum _{i=1}^{n}((b_{f,i}+b_{g,i}){\mbox{sen}}(ix)+(c_{f,i}+c_{g,i})\cos(ix))\in L^{2}}"></span>,</dd></dl></dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (af)(x):=af(x)_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>a</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (af)(x):=af(x)_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8954ab66432b00aa9f25a62ecc93606710e7365" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.849ex; height:2.843ex;" alt="{\displaystyle (af)(x):=af(x)_{}^{}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =a(a_{f}\sum _{i=1}^{n}(b_{f,i}{\mbox{sen}}(ix)+c_{f,i}\cos(ix)))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>sen</mtext> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =a(a_{f}\sum _{i=1}^{n}(b_{f,i}{\mbox{sen}}(ix)+c_{f,i}\cos(ix)))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b63dd5b9d5d98e6d091c93366e41a3b58b6697" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.199ex; height:6.843ex;" alt="{\displaystyle =a(a_{f}\sum _{i=1}^{n}(b_{f,i}{\mbox{sen}}(ix)+c_{f,i}\cos(ix)))}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =aa_{f}\sum _{i=1}^{n}(ab_{f,i}{\mbox{sen}}(ix)+ac_{f,i}\cos(ix))\in L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>a</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>a</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>sen</mtext> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>a</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =aa_{f}\sum _{i=1}^{n}(ab_{f,i}{\mbox{sen}}(ix)+ac_{f,i}\cos(ix))\in L^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6245620787ca6d02b3f2581071737e0667e21ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.327ex; height:6.843ex;" alt="{\displaystyle =aa_{f}\sum _{i=1}^{n}(ab_{f,i}{\mbox{sen}}(ix)+ac_{f,i}\cos(ix))\in L^{2}}"></span>.</dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Los_sistemas_de_ecuaciones_lineales_homogéneas"><span id="Los_sistemas_de_ecuaciones_lineales_homog.C3.A9neas"></span>Los sistemas de ecuaciones lineales homogéneas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=13" title="Editar sección: Los sistemas de ecuaciones lineales homogéneas"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint AP rellink"><span style="font-size:88%">Artículos principales:</span> <i><a href="/wiki/Ecuaci%C3%B3n_lineal" class="mw-redirect" title="Ecuación lineal"> Ecuación lineal</a></i><span style="font-size:88%">, </span><i><a href="/wiki/Ecuaci%C3%B3n_diferencial_lineal" title="Ecuación diferencial lineal"> Ecuación diferencial lineal</a></i><span style="font-size:88%"> y </span><i><a href="/wiki/Sistemas_de_ecuaciones_lineales" class="mw-redirect" title="Sistemas de ecuaciones lineales"> Sistemas de ecuaciones lineales</a></i>.</div> <figure typeof="mw:File/Thumb"><a href="/wiki/Archivo:Intersecci%C3%B3nEspacioVectorial.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Intersecci%C3%B3nEspacioVectorial.gif/170px-Intersecci%C3%B3nEspacioVectorial.gif" decoding="async" width="170" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Intersecci%C3%B3nEspacioVectorial.gif/255px-Intersecci%C3%B3nEspacioVectorial.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Intersecci%C3%B3nEspacioVectorial.gif/340px-Intersecci%C3%B3nEspacioVectorial.gif 2x" data-file-width="417" data-file-height="500" /></a><figcaption>Sistema de 2 ecuaciones y 3 variables</figcaption></figure> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\begin{matrix}a_{1,1}x_{1}&+\dots &+a_{1,n}x_{n}&=0\\\vdots &&\vdots &\vdots \\a_{m,1}x_{1}&+\dots &+a_{m,n}x_{n}&=0\end{matrix}}\end{cases}}\;\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>+</mo> <mo>…</mo> </mtd> <mtd> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>+</mo> <mo>…</mo> </mtd> <mtd> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\begin{matrix}a_{1,1}x_{1}&+\dots &+a_{1,n}x_{n}&=0\\\vdots &&\vdots &\vdots \\a_{m,1}x_{1}&+\dots &+a_{m,n}x_{n}&=0\end{matrix}}\end{cases}}\;\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369b4eb2af50ec2871d69db51e4c74141ecbcf78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.62ex; margin-bottom: -0.218ex; width:35.634ex; height:10.843ex;" alt="{\displaystyle {\begin{cases}{\begin{matrix}a_{1,1}x_{1}&+\dots &+a_{1,n}x_{n}&=0\\\vdots &&\vdots &\vdots \\a_{m,1}x_{1}&+\dots &+a_{m,n}x_{n}&=0\end{matrix}}\end{cases}}\;\;}"></span> o equivalentemente <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}a_{1,1}&+\dots &+a_{1,n}\\\vdots &&\vdots &\\a_{m,1}&+\dots &+a_{m,n}\end{pmatrix}}{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}={\begin{pmatrix}0\\\vdots \\0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>+</mo> <mo>…</mo> </mtd> <mtd> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> <mtd /> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>+</mo> <mo>…</mo> </mtd> <mtd> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}a_{1,1}&+\dots &+a_{1,n}\\\vdots &&\vdots &\\a_{m,1}&+\dots &+a_{m,n}\end{pmatrix}}{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}={\begin{pmatrix}0\\\vdots \\0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ca835faf1d2033507ab1cb06f3c5477d7bb3d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.62ex; margin-bottom: -0.218ex; width:43.491ex; height:10.843ex;" alt="{\displaystyle {\begin{pmatrix}a_{1,1}&+\dots &+a_{1,n}\\\vdots &&\vdots &\\a_{m,1}&+\dots &+a_{m,n}\end{pmatrix}}{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}={\begin{pmatrix}0\\\vdots \\0\end{pmatrix}}}"></span> simplificado como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{}^{}x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{}^{}x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd971b6fecf303a8751a79166df06bc35038ddfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.334ex; height:2.176ex;" alt="{\displaystyle A_{}^{}x=0}"></span> </p><p>Un sistema de ecuaciones lineales homogéneas( ecuaciones lineales en las que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msubsup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c399dad14ebba411d1c15f220933224e0e2acff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0_{}^{}}"></span> es siempre una solución, es decir, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\;\dots ,\;x_{n})=(0,\;\dots ,\;0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},\;\dots ,\;x_{n})=(0,\;\dots ,\;0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c46e29e9c040ee7b2f0ac8292ee042b0e54580d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.911ex; height:2.843ex;" alt="{\displaystyle (x_{1},\;\dots ,\;x_{n})=(0,\;\dots ,\;0)}"></span>) posee soluciones que forman un espacio vectorial, se puede ver en sus dos operaciones: </p> <dl><dd>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax=0,Ay=0\Rightarrow Ax+Ay=0\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>A</mi> <mi>y</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>A</mi> <mi>y</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax=0,Ay=0\Rightarrow Ax+Ay=0\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1816a4d72cacdddd2b28be593d111dceaa2ed4b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.183ex; height:2.509ex;" alt="{\displaystyle Ax=0,Ay=0\Rightarrow Ax+Ay=0\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(x+y)=0_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(x+y)=0_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54f53056170dfb078ec60b334e4db668c9b43cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.139ex; height:2.843ex;" alt="{\displaystyle A(x+y)=0_{}^{}}"></span></dd></dl> <dl><dd>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax=0,a\in K\Rightarrow a(Ax)=0\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mo>∈</mo> <mi>K</mi> <mo stretchy="false">⇒</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax=0,a\in K\Rightarrow a(Ax)=0\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39b51cccf3aded72fa0a6e903d60840219a03c47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.46ex; height:2.843ex;" alt="{\displaystyle Ax=0,a\in K\Rightarrow a(Ax)=0\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(ax)=0_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(ax)=0_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1892b679cb67b74ebfc63ddf97ba6a499afd58b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.373ex; height:2.843ex;" alt="{\displaystyle A(ax)=0_{}^{}}"></span>.</dd></dl> <p>También que las ecuaciones en sí, filas de la matriz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5b1ae220b9d439abd63542ea83249d7701b8a8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A_{}^{}}"></span> notadas como una matriz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bce5f6a6d0d32834484048c16f3b39f9c23d076" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle 1\times n}"></span>, es decir, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}=(a_{i,1},\;\dots ,\;a_{i,n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}=(a_{i,1},\;\dots ,\;a_{i,n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fa292467026af59d81a8120a5b8670d0917bf63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.673ex; height:3.009ex;" alt="{\displaystyle E_{i}=(a_{i,1},\;\dots ,\;a_{i,n})}"></span>, son un espacio vectorial, como se puede ver en sus dos operaciones: </p> <dl><dd>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}x=0,\;E_{j}x=0\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}x=0,\;E_{j}x=0\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/913f9bfce9cb6c1bbf144cdd05690a5823f3bfe0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.969ex; height:2.843ex;" alt="{\displaystyle E_{i}x=0,\;E_{j}x=0\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}^{}x+E_{j}x=0\Rightarrow (E_{i}+E_{j})x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mi>x</mi> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}^{}x+E_{j}x=0\Rightarrow (E_{i}+E_{j})x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cfa622710f8a533c3a9bbc34be18038fbcfa080" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.895ex; height:3.009ex;" alt="{\displaystyle E_{i}^{}x+E_{j}x=0\Rightarrow (E_{i}+E_{j})x=0}"></span></dd></dl> <dl><dd>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}x=0,\;a\in K\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>a</mi> <mo>∈</mo> <mi>K</mi> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}x=0,\;a\in K\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b35d6ea895e11d3e7980a89ba9bcf15995cc9062" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.89ex; height:2.509ex;" alt="{\displaystyle E_{i}x=0,\;a\in K\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(E_{i}^{}x)=0\Rightarrow (aE_{i})x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <mi>a</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(E_{i}^{}x)=0\Rightarrow (aE_{i})x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77e5a2101160cac8dcaba8e96a832c8e972894e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.903ex; height:2.843ex;" alt="{\displaystyle a(E_{i}^{}x)=0\Rightarrow (aE_{i})x=0}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Subespacio_vectorial">Subespacio vectorial</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=14" title="Editar sección: Subespacio vectorial"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Definición_2"><span id="Definici.C3.B3n_2"></span>Definición</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=15" title="Editar sección: Definición"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sea <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> un espacio vectorial sobre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⊆</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\subseteq V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7720f1387a2e51d58daf1fb9e9b1b730430b8466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.668ex; height:2.343ex;" alt="{\displaystyle U\subseteq V}"></span> un <a href="/wiki/Subconjunto" title="Subconjunto">subconjunto</a> no vacío de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, se dice que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> es un <a href="/wiki/Subespacio_vectorial" title="Subespacio vectorial">subespacio vectorial</a> de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> si: </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u+v\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u+v\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ddae2e87ec0d250f8c24d67c4d393cc4a7bc578" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.921ex; height:2.343ex;" alt="{\displaystyle u+v\in U}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta u\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mi>u</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta u\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9af7188c7b7bbe8782a3f6c2ab0901706a64696" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.285ex; height:2.509ex;" alt="{\displaystyle \beta u\in U}"></span></li></ol> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \;u,v\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \;u,v\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b5914e69796838edf7f61c129cd53fe79d8f7a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.052ex; height:2.509ex;" alt="{\displaystyle \forall \;u,v\in U}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc3d1cd74617fc6eb6f78b80c28a40319bed8abf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.239ex; height:2.509ex;" alt="{\displaystyle \beta \in K}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Consecuencias">Consecuencias</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=16" title="Editar sección: Consecuencias"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> hereda las operaciones de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> como aplicaciones bien definidas, es decir que no escapan de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>, y como consecuencia tenemos que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> es un espacio vectorial sobre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>. </p><p>Con cualquier subconjunto de elementos seleccionados en los espacios vectoriales anteriores, no vacío, se pueden generar subespacios vectoriales, para ello sería útil introducir nuevos conceptos que facilitarán el trabajo sobre estos nuevos espacios vectoriales. </p> <div class="mw-heading mw-heading2"><h2 id="Resultados_internos">Resultados internos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=17" title="Editar sección: Resultados internos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Para detallar el comportamiento interno de todos los espacios vectoriales de modo general es necesario exponer una serie de herramientas cronológicamente vinculadas entre ellas, con las cuales es posible construir resultados válidos en cualquier estructura que sea espacio vectorial. </p> <div class="mw-heading mw-heading3"><h3 id="Combinación_lineal"><span id="Combinaci.C3.B3n_lineal"></span>Combinación lineal</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=18" title="Editar sección: Combinación lineal"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Archivo:VectorGenerado.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/VectorGenerado.gif/250px-VectorGenerado.gif" decoding="async" width="250" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/VectorGenerado.gif/375px-VectorGenerado.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/9/95/VectorGenerado.gif 2x" data-file-width="472" data-file-height="306" /></a><figcaption>Cada vector <i>u</i> es combinación lineal de forma única</figcaption></figure> <p>Dado un espacio vectorial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28f93f365b478d2995b9c06f677212f50382aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E_{}^{}}"></span>, diremos que un vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∈</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\in E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf0244d21924fc6cf6f050679b6254db00a31631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.946ex; height:2.176ex;" alt="{\displaystyle u\in E}"></span> es <b><a href="/wiki/Combinaci%C3%B3n_lineal" title="Combinación lineal">combinación lineal</a></b> de los vectores de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\{v_{1},\dots ,v_{n}\}\subseteq E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>⊆</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\{v_{1},\dots ,v_{n}\}\subseteq E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a10f80b9c4e5b221e7a208cf3a8d96171809b02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.503ex; height:2.843ex;" alt="{\displaystyle S=\{v_{1},\dots ,v_{n}\}\subseteq E}"></span> si existen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\dots ,a_{n}\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\dots ,a_{n}\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb29fafa8cccffc7ef0b8bfbd89c757ddd2d8b62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.429ex; height:2.509ex;" alt="{\displaystyle a_{1},\dots ,a_{n}\in \mathbb {R} }"></span> tales que </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=a_{1}v_{1}+\cdots +a_{n}v_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=a_{1}v_{1}+\cdots +a_{n}v_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a9550a1ff9e91a690b151f8771800ea371658a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.092ex; height:2.343ex;" alt="{\displaystyle u=a_{1}v_{1}+\cdots +a_{n}v_{n}}"></span> </p> </blockquote> <p>Denotaremos como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle S_{}^{}\rangle _{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle S_{}^{}\rangle _{E}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17943c1614ac71aeaf7a504a85c7296a79d6b8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.796ex; height:2.843ex;" alt="{\displaystyle \langle S_{}^{}\rangle _{E}}"></span> el conjunto resultante de todas las combinaciones lineales de los vectores de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{}^{}\subset E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo>⊂</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{}^{}\subset E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71d7cb56ccea14fa837f0a3f09b93ba67eebe675" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.373ex; height:2.176ex;" alt="{\displaystyle S_{}^{}\subset E}"></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Proposición_1"><span id="Proposici.C3.B3n_1"></span>Proposición 1</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=19" title="Editar sección: Proposición 1"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dado <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28f93f365b478d2995b9c06f677212f50382aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E_{}^{}}"></span> un espacio vectorial y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subset E_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊂</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subset E_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39799a60bb136ba7c07b287a0f53cba13510f882" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.373ex; height:2.176ex;" alt="{\displaystyle S\subset E_{}^{}}"></span> un conjunto de vectores, el conjunto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=\langle S_{}^{}\rangle _{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=\langle S_{}^{}\rangle _{E}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5431de7b70554653d36f76032bcdd18fa7f60761" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.636ex; height:2.843ex;" alt="{\displaystyle F=\langle S_{}^{}\rangle _{E}}"></span> es el subespacio vectorial más pequeño contenido en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28f93f365b478d2995b9c06f677212f50382aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E_{}^{}}"></span> y que contiene a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82dcee00b5702592ec80c871c9ec1af4039c799b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S_{}^{}}"></span>. </p> <table class="mw-collapsible wikitable mw-collapsed" width="75%" style="text-align:left; padding:0px; background-color: var(--background-color-neutral-subtle, #f9f9f9); color: var(--color-base, #202122);"> <tbody><tr> <td style="text-align:left; font-weight:bold;">Demostración </td></tr> <tr> <td style="background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align:left; font-size:95%; padding:6px"> <p>Si se supone lo contrario, que existe uno más pequeño <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{}^{}\varsubsetneq F\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo class="MJX-variant">⊊</mo> <mi>F</mi> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{}^{}\varsubsetneq F\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a68d19248d69560b7ff81ab7c603cbcb65d69b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.635ex; height:2.676ex;" alt="{\displaystyle G_{}^{}\varsubsetneq F\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists u\in F:u\notin G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>u</mi> <mo>∈</mo> <mi>F</mi> <mo>:</mo> <mi>u</mi> <mo>∉</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists u\in F:u\notin G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c229fda414a7735658f9d24b10dad7168be2692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.138ex; height:2.676ex;" alt="{\displaystyle \exists u\in F:u\notin G}"></span> contradicción, ya que <i>u</i> está generado por elementos de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subset F\Rightarrow u\in G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊂</mo> <mi>F</mi> <mo stretchy="false">⇒</mo> <mi>u</mi> <mo>∈</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subset F\Rightarrow u\in G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b448cec5c2e1c2ea88844d3b40f4621cac8e128a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.95ex; height:2.176ex;" alt="{\displaystyle S\subset F\Rightarrow u\in G}"></span> a causa de la buena definición de las dos operaciones, por tanto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=G_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=G_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcf3dc79db3cec4eb84d3495968065ced33f6670" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.666ex; height:2.176ex;" alt="{\displaystyle F=G_{}^{}}"></span>. </p> </td></tr></tbody></table> <dl><dd><b>Nota</b>. En este caso se dice que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82dcee00b5702592ec80c871c9ec1af4039c799b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S_{}^{}}"></span> es un <b><a href="/wiki/Sistema_generador" title="Sistema generador">sistema de generadores</a></b> que genera a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba69787528be36f0312673a5ffe86596b55f4a24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F_{}^{}}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Independencia_lineal">Independencia lineal</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=20" title="Editar sección: Independencia lineal"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Diremos que un conjunto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{}^{}=\{v_{1},\;\dots ,\;v_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{}^{}=\{v_{1},\;\dots ,\;v_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f008de883e23bd6aab338c6de943e3fdd18741c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.919ex; height:2.843ex;" alt="{\displaystyle S_{}^{}=\{v_{1},\;\dots ,\;v_{n}\}}"></span> de vectores es <b><a href="/wiki/Independencia_lineal" class="mw-redirect" title="Independencia lineal">linealmente independiente</a></b> si el vector 0 no se puede expresar como combinación lineal no nula de los vectores de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82dcee00b5702592ec80c871c9ec1af4039c799b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S_{}^{}}"></span>, es decir: </p> <dl><dd>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=a_{1}v_{1}+\cdots +a_{n}v_{n}\Rightarrow a_{1}=\cdots =a_{n}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">⇒</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>⋯</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=a_{1}v_{1}+\cdots +a_{n}v_{n}\Rightarrow a_{1}=\cdots =a_{n}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0ca872baadd2c76be0cd8eaa7d1033274e3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:43.453ex; height:2.509ex;" alt="{\displaystyle 0=a_{1}v_{1}+\cdots +a_{n}v_{n}\Rightarrow a_{1}=\cdots =a_{n}=0}"></span>.</dd></dl> <p>Diremos que un conjunto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82dcee00b5702592ec80c871c9ec1af4039c799b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S_{}^{}}"></span> de vectores es <b>linealmente dependiente</b> si no es linealmente independiente. </p> <div class="mw-heading mw-heading4"><h4 id="Proposición_2"><span id="Proposici.C3.B3n_2"></span>Proposición 2</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=21" title="Editar sección: Proposición 2"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1},\;\dots ,\;v_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1},\;\dots ,\;v_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc40f02a4227ed1e450bd5789e4c685abed4eaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.996ex; height:2.009ex;" alt="{\displaystyle v_{1},\;\dots ,\;v_{n}}"></span> son linealmente dependientes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow \exists v_{i}\neq 0:v_{i}=\sum _{i\neq j\geq 1}^{n}a_{j}v_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> <mi mathvariant="normal">∃</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> <mo>:</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>≠</mo> <mi>j</mi> <mo>≥</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow \exists v_{i}\neq 0:v_{i}=\sum _{i\neq j\geq 1}^{n}a_{j}v_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/323f310476a52a3303d1a7d123776c2af16afc64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:26.6ex; height:7.343ex;" alt="{\displaystyle \Leftrightarrow \exists v_{i}\neq 0:v_{i}=\sum _{i\neq j\geq 1}^{n}a_{j}v_{j}}"></span> </p> <table class="mw-collapsible wikitable mw-collapsed" width="75%" style="text-align:left; padding:0px; background-color: var(--background-color-neutral-subtle, #f9f9f9); color: var(--color-base, #202122);"> <tbody><tr> <td style="text-align:left; font-weight:bold;">Demostración </td></tr> <tr> <td style="background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align:left; font-size:95%; padding:6px"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04e2b489988c51eb309422922ad41fc9b36223b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.228ex; height:2.843ex;" alt="{\displaystyle \Rightarrow )}"></span> Linealmente dependientes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow 0=b_{1}v_{1}+\cdots +b_{n}v_{n}:\exists b_{i}\neq 0\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mn>0</mn> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mi mathvariant="normal">∃</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow 0=b_{1}v_{1}+\cdots +b_{n}v_{n}:\exists b_{i}\neq 0\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1763fda547427642123edc0adab797b34b93dd23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.686ex; height:2.676ex;" alt="{\displaystyle \Rightarrow 0=b_{1}v_{1}+\cdots +b_{n}v_{n}:\exists b_{i}\neq 0\Rightarrow }"></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{i}v_{i}=-\sum _{i\neq j\geq 1}^{n}b_{j}v_{j}\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>≠</mo> <mi>j</mi> <mo>≥</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{i}v_{i}=-\sum _{i\neq j\geq 1}^{n}b_{j}v_{j}\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/708f0b560d43041585b109654c9a45f4a8d4a560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:20.943ex; height:7.343ex;" alt="{\displaystyle b_{i}v_{i}=-\sum _{i\neq j\geq 1}^{n}b_{j}v_{j}\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}=\sum _{i\neq j\geq 1}^{n}(-b_{j}b_{i}^{-1})v_{j}=\sum _{i\neq j\geq 1}^{n}a_{j}v_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>≠</mo> <mi>j</mi> <mo>≥</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>≠</mo> <mi>j</mi> <mo>≥</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}=\sum _{i\neq j\geq 1}^{n}(-b_{j}b_{i}^{-1})v_{j}=\sum _{i\neq j\geq 1}^{n}a_{j}v_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17ada81cd64021f4efc0eb1213d7bc50ca46bea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:32.828ex; height:7.343ex;" alt="{\displaystyle v_{i}=\sum _{i\neq j\geq 1}^{n}(-b_{j}b_{i}^{-1})v_{j}=\sum _{i\neq j\geq 1}^{n}a_{j}v_{j}}"></span> tomando <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{j}=-b_{j}b_{i}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{j}=-b_{j}b_{i}^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01066468704b263f6c016daac89caba6289c6e49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.284ex; height:3.343ex;" alt="{\displaystyle a_{j}=-b_{j}b_{i}^{-1}}"></span>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftarrow )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇐</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftarrow )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fb3a195279c2030e360a943ee5644d37a5ac085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.228ex; height:2.843ex;" alt="{\displaystyle \Leftarrow )}"></span> Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}=\sum _{i\neq j\geq 1}^{n}a_{j}v_{j}\Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>≠</mo> <mi>j</mi> <mo>≥</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}=\sum _{i\neq j\geq 1}^{n}a_{j}v_{j}\Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/569fda6bc2d6dd0a5a35f98a01dbf18e6630b87c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:17.183ex; height:7.343ex;" alt="{\displaystyle v_{i}=\sum _{i\neq j\geq 1}^{n}a_{j}v_{j}\Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=a_{1}v_{1}+\cdots +a_{n}v_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=a_{1}v_{1}+\cdots +a_{n}v_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/940517061cec22aa6882b872426aa57fbeb6d7dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.925ex; height:2.509ex;" alt="{\displaystyle 0=a_{1}v_{1}+\cdots +a_{n}v_{n}}"></span> donde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}:=-1\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:=</mo> <mo>−</mo> <mn>1</mn> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}:=-1\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caaaf91a27b6f02a2eeae296a861f12a2a4e86a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.006ex; height:2.676ex;" alt="{\displaystyle a_{i}:=-1\neq 0}"></span> y por tanto linealmente dependientes. </p> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Base_de_un_espacio_vectorial">Base de un espacio vectorial</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=22" title="Editar sección: Base de un espacio vectorial"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint AP rellink"><span style="font-size:88%">Artículos principales:</span> <i><a href="/wiki/Base_(%C3%A1lgebra)" title="Base (álgebra)"> Base</a></i><span style="font-size:88%"> y </span><i><a href="/wiki/Dimensi%C3%B3n_de_un_espacio_vectorial" title="Dimensión de un espacio vectorial"> Dimensión</a></i>.</div> <p>Las <cite id="label1"><i>bases</i></cite> revelan la estructura de los espacios vectoriales de una manera concisa. Una base es el menor conjunto (finito o infinito) <span style="white-space:nowrap"><i>B</i> = {<b>v</b><sub><i>i</i></sub>}<sub><i>i</i> ∈ <i>I</i></sub></span> de vectores que generan todo el espacio. Esto significa que cualquier vector <b>v</b> puede ser expresado como una suma (llamada <i><a href="/wiki/Combinaci%C3%B3n_lineal" title="Combinación lineal">combinación lineal</a></i>) de elementos de la base </p> <dl><dd><i>a</i><sub>1</sub><b>v</b><sub><i>i</i><sub>1</sub></sub> + <i>a</i><sub>2</sub><b>v</b><sub><i>i</i><sub>2</sub></sub> + ... + <i>a</i><sub><i>n</i></sub><b>v</b><sub><i>i</i><sub><i>n</i></sub></sub>,</dd></dl> <p>donde los <i>a</i><sub><i>k</i></sub> son escalares y <b>v</b><sub><i>i</i><sub><i>k</i></sub></sub> <span style="white-space:nowrap">(<i>k</i> = 1, ..., <i>n</i>)</span> elementos de la base <i>B</i>. La minimalidad, por otro lado, se hace formal por el concepto de <a href="/wiki/Independencia_lineal" class="mw-redirect" title="Independencia lineal">independencia lineal</a>. Un conjunto de vectores se dice que es linealmente independiente si ninguno de sus elementos puede ser expresado como una combinación lineal de los restantes. Equivalentemente, una ecuación </p> <dl><dd><i>a</i><sub>1</sub><b>v</b><sub><i>i</i><sub>1</sub></sub> + <i>a</i><sub><i>i</i><sub>2</sub></sub><b>v</b><sub>2</sub> + ... + <i>a</i><sub><i>n</i></sub><b>v</b><sub><i>i</i><sub>n</sub></sub> = 0</dd></dl> <p>solo se consigue si todos los escalares <i>a</i><sub>1</sub>, ..., <i>a</i><sub><i>n</i></sub> son iguales a cero. Por definición de la base cada vector puede ser expresado como una suma finita de los elementos de la base. Debido a la independencia lineal este tipo de representación es única. Los espacios vectoriales a veces se introducen desde este punto de vista. </p> <div class="mw-heading mw-heading3"><h3 id="Base_formalmente">Base formalmente</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=23" title="Editar sección: Base formalmente"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Archivo:BaseGeneradora.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/BaseGeneradora.gif/200px-BaseGeneradora.gif" decoding="async" width="200" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/BaseGeneradora.gif/300px-BaseGeneradora.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/c/c9/BaseGeneradora.gif 2x" data-file-width="394" data-file-height="288" /></a><figcaption>v<sub>1</sub> y v<sub>2</sub> son base de un plano, si hubiese dependencia lineal (alineados), la cuadrícula no podría generarse.</figcaption></figure> <p>Dado un sistema de generadores, diremos que es una <b><a href="/wiki/Base_(%C3%A1lgebra)" title="Base (álgebra)">base</a></b> si son linealmente independientes. </p> <dl><dd><b>Proposición 3.</b> Dado un espacio vectorial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E,\;\{v_{1},\;\dots ,v_{n}\}=F\subset E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mo fence="false" stretchy="false">{</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>F</mi> <mo>⊂</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E,\;\{v_{1},\;\dots ,v_{n}\}=F\subset E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094768acc55cd44ed9ff03b9ad5b5ea4308d7f3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.844ex; height:2.843ex;" alt="{\displaystyle E,\;\{v_{1},\;\dots ,v_{n}\}=F\subset E}"></span> es una base <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall u\in E,\;\exists !a_{i}\in K,\;i\in {1,\;\dots ,n}:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>u</mi> <mo>∈</mo> <mi>E</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃</mi> <mo>!</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>K</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mrow> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall u\in E,\;\exists !a_{i}\in K,\;i\in {1,\;\dots ,n}:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40f62c6a28850ee3fb5cf1dcddd8af447e4c5844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.788ex; height:2.509ex;" alt="{\displaystyle \forall u\in E,\;\exists !a_{i}\in K,\;i\in {1,\;\dots ,n}:}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=\sum _{i=1}^{n}a_{i}v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=\sum _{i=1}^{n}a_{i}v_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fcd6f699567a1d37b47140b53caa442d80eabf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.127ex; height:6.843ex;" alt="{\displaystyle u=\sum _{i=1}^{n}a_{i}v_{i}}"></span>.</dd></dl> <dl><dd><b>Proposición 4.</b> Dado un espacio vectorial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E,\;S=\{v_{1},\;\dots ,\;v_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E,\;S=\{v_{1},\;\dots ,\;v_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/053b2b7edafc8801fbb4826e6ac47ac905cd45d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.374ex; height:2.843ex;" alt="{\displaystyle E,\;S=\{v_{1},\;\dots ,\;v_{n}\}}"></span> linealmente independiente y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\notin \langle S\rangle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∉</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>S</mi> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\notin \langle S\rangle \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2a3e7c1112ec57a7c40651f97731b3568e24fe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.448ex; height:2.843ex;" alt="{\displaystyle u\notin \langle S\rangle \Rightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{u\}\cup S=\{u,\;v_{1},\;\dots ,\;v_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>u</mi> <mo fence="false" stretchy="false">}</mo> <mo>∪</mo> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>u</mi> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{u\}\cup S=\{u,\;v_{1},\;\dots ,\;v_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7566f3e7f3d5d82acce523657cfa406749d6322" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.165ex; height:2.843ex;" alt="{\displaystyle \{u\}\cup S=\{u,\;v_{1},\;\dots ,\;v_{n}\}}"></span> son linealmente independiente.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Teorema_de_la_base_de_generadores">Teorema de la base de generadores</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=24" title="Editar sección: Teorema de la base de generadores"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Todo sistema de generadores tiene una base. </p> <div class="mw-heading mw-heading4"><h4 id="Teorema_Steinitz">Teorema Steinitz</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=25" title="Editar sección: Teorema Steinitz"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Toda base de un espacio vectorial puede ser cambiada parcialmente por vectores linealmente independientes. </p> <dl><dd><b>Corolario</b>. Si un espacio vectorial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28f93f365b478d2995b9c06f677212f50382aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E_{}^{}}"></span> tiene una base de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81ddf8a14e800a627ca8a06f7249a12c670d024a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n_{}^{}}"></span> vectores <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"></span> cualquier otra base posee <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81ddf8a14e800a627ca8a06f7249a12c670d024a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n_{}^{}}"></span> vectores.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Observación"><span id="Observaci.C3.B3n"></span>Observación</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=26" title="Editar sección: Observación"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Todo espacio vectorial</b> tiene una base. Este hecho se basa en el <a href="/wiki/Lema_de_Zorn" title="Lema de Zorn">lema de Zorn</a>, una formulación equivalente del <a href="/wiki/Axioma_de_elecci%C3%B3n" title="Axioma de elección">axioma de elección</a>. Habida cuenta de los otros axiomas de la <a href="/wiki/Teor%C3%ADa_de_conjuntos_de_Zermelo-Fraenkel" class="mw-redirect" title="Teoría de conjuntos de Zermelo-Fraenkel">teoría de conjuntos de Zermelo-Fraenkel</a>, la existencia de bases es equivalente al axioma de elección. El <a href="/w/index.php?title=Ultrafilter_lemma&action=edit&redlink=1" class="new" title="Ultrafilter lemma (aún no redactado)">ultrafilter lemma</a>, que es más débil que el axioma de elección, implica que todas las bases de un espacio vectorial tienen el mismo "tamaño", es decir, <a href="/wiki/Cardinalidad" title="Cardinalidad">cardinalidad</a>. Si el espacio es generado por un número finito de vectores, todo lo anterior puede demostrarse sin necesidad de acudir a la teoría de conjuntos. </p> <div class="mw-heading mw-heading3"><h3 id="Dimensión"><span id="Dimensi.C3.B3n"></span>Dimensión</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=27" title="Editar sección: Dimensión"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dado un espacio vectorial sobre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a6c0f78592eaf8c5b477d9551ea3adffeea33fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K_{}^{}}"></span>: </p> <ul><li>Si tiene base finita, diremos <b><a href="/wiki/Dimensi%C3%B3n_de_un_espacio_vectorial" title="Dimensión de un espacio vectorial">dimensión</a></b> al número de elementos de dicha base.</li> <li>Si tiene base no finita, diremos que es de <b><a href="/wiki/Dimensi%C3%B3n_de_un_espacio_vectorial" title="Dimensión de un espacio vectorial">dimensión infinita</a></b>.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Notación_2"><span id="Notaci.C3.B3n_2"></span>Notación</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=28" title="Editar sección: Notación"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dado un espacio vectorial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28f93f365b478d2995b9c06f677212f50382aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E_{}^{}}"></span> y un subespacio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{}^{}\subset E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo>⊂</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{}^{}\subset E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa4c6b4702536c5003aa24378c67a834151e27d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.615ex; height:2.176ex;" alt="{\displaystyle F_{}^{}\subset E}"></span>, tenemos que: </p> <ul><li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28f93f365b478d2995b9c06f677212f50382aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E_{}^{}}"></span> tiene dimensión <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81ddf8a14e800a627ca8a06f7249a12c670d024a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n_{}^{}}"></span> lo indicaremos como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim(E)=n_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(E)=n_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e11c2452474717689f4e8bd543c0e7ed284268d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.953ex; height:2.843ex;" alt="{\displaystyle \dim(E)=n_{}^{}}"></span>.</li> <li>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba69787528be36f0312673a5ffe86596b55f4a24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F_{}^{}}"></span> tiene dimensión <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/409c666fc7643f79add112796e3624f79064bcfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m_{}^{}}"></span> como subespacio de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28f93f365b478d2995b9c06f677212f50382aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E_{}^{}}"></span> lo indicaremos como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim _{E}(F)=m_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>dim</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim _{E}(F)=m_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22ea5664c1c123f11b6ab0282e34cd4d04ff773f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.052ex; height:2.843ex;" alt="{\displaystyle \dim _{E}(F)=m_{}^{}}"></span>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Intersección_de_subespacios_vectoriales"><span id="Intersecci.C3.B3n_de_subespacios_vectoriales"></span>Intersección de subespacios vectoriales</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=29" title="Editar sección: Intersección de subespacios vectoriales"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dado dos subespacios vectoriales <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F,G\subset E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo>⊂</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F,G\subset E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ce13a3263efe8cc2a27699dda2cc869be6fc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.476ex; height:2.509ex;" alt="{\displaystyle F,G\subset E}"></span>, la intersección es subespacio vectorial contenido en estos y lo notaremos como: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\cap G:=\{u:\;u\in F,\;u\in G\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>∩</mo> <mi>G</mi> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>u</mi> <mo>:</mo> <mspace width="thickmathspace" /> <mi>u</mi> <mo>∈</mo> <mi>F</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>u</mi> <mo>∈</mo> <mi>G</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\cap G:=\{u:\;u\in F,\;u\in G\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5015715166bd03025b00c80f21e14b19a3708144" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.72ex; height:2.843ex;" alt="{\displaystyle F\cap G:=\{u:\;u\in F,\;u\in G\}}"></span>.</dd></dl> <dl><dd><b>Observaciones</b>. Para la intersección sucesiva de espacios vectoriales se procede, inductivamente, de dos en dos.</dd></dl> <p>La unión de subespacios vectoriales no es en general un subespacio vectorial. </p> <div class="mw-heading mw-heading3"><h3 id="Suma_de_subespacios_vectoriales">Suma de subespacios vectoriales</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=30" title="Editar sección: Suma de subespacios vectoriales"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dado dos subespacios vectoriales <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F,G\subset E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo>⊂</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F,G\subset E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ce13a3263efe8cc2a27699dda2cc869be6fc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.476ex; height:2.509ex;" alt="{\displaystyle F,G\subset E}"></span>, la suma es un subespacio vectorial que contiene a estos y la notaremos como: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F+G:=\{u=v_{1}+v_{2}:\;v_{1}\in F,\;v_{2}\in G\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>+</mo> <mi>G</mi> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>u</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mspace width="thickmathspace" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mi>F</mi> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mi>G</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F+G:=\{u=v_{1}+v_{2}:\;v_{1}\in F,\;v_{2}\in G\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205cfc5d371f8620e3f86c4c4a2a6b9773401e80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.984ex; height:2.843ex;" alt="{\displaystyle F+G:=\{u=v_{1}+v_{2}:\;v_{1}\in F,\;v_{2}\in G\}}"></span>.</dd></dl> <p>Si F y G son subespacios vectoriales de E, su suma F+G es el subespacio vectorial de E más pequeño que contiene a F y a G. </p> <dl><dd><b>Observación</b>. Para la suma sucesiva de espacios vectoriales se procede, inductivamente, de dos en dos.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Teorema_Fórmula_de_Grassmann"><span id="Teorema_F.C3.B3rmula_de_Grassmann"></span>Teorema Fórmula de Grassmann</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=31" title="Editar sección: Teorema Fórmula de Grassmann"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dado dos subespacios vectoriales <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F,G\subset E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo>⊂</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F,G\subset E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ce13a3263efe8cc2a27699dda2cc869be6fc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.476ex; height:2.509ex;" alt="{\displaystyle F,G\subset E}"></span> de dimensión finita, tenemos el resultado siguiente: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim _{E}(F+G)=\dim _{E}(F)+\dim _{E}(G)-\dim _{E}(F\cap G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>dim</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo>+</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>dim</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>dim</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>dim</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo>∩</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim _{E}(F+G)=\dim _{E}(F)+\dim _{E}(G)-\dim _{E}(F\cap G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a5c88f9dd5cc6dae09f1fbe82c61c496752d9fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.594ex; height:2.843ex;" alt="{\displaystyle \dim _{E}(F+G)=\dim _{E}(F)+\dim _{E}(G)-\dim _{E}(F\cap G)}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Suma_directa_de_subespacios_vectoriales">Suma directa de subespacios vectoriales</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=32" title="Editar sección: Suma directa de subespacios vectoriales"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dados dos subespacios vectoriales <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F,G\subset E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo>⊂</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F,G\subset E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ce13a3263efe8cc2a27699dda2cc869be6fc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.476ex; height:2.509ex;" alt="{\displaystyle F,G\subset E}"></span>, diremos que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F+G_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>+</mo> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F+G_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/758b30f19c983ec6b1779ddea143d26849819ddb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.408ex; height:2.343ex;" alt="{\displaystyle F+G_{}^{}}"></span> es una <i>suma directa</i> si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\cap G={0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>∩</mo> <mi>G</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\cap G={0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24167af0188d8676dbfde653b6edc3bb8d813b3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.411ex; height:2.176ex;" alt="{\displaystyle F\cap G={0}}"></span> y lo denotaremos como: </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\oplus G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>⊕</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\oplus G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35068f507f715f0328d1fa39cb6202280684f646" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.408ex; height:2.343ex;" alt="{\displaystyle F\oplus G}"></span>.</dd></dl></dd></dl> <p>Cuando <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> están en suma directa, cada vector de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F+G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>+</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F+G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbdfe6e6306ae2363bd55c55e11a4341a3fbe98a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.408ex; height:2.343ex;" alt="{\displaystyle F+G}"></span> se expresa de forma única como suma de un vector de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> y otro vector de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Cociente_de_espacios_vectoriales">Cociente de espacios vectoriales</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=33" title="Editar sección: Cociente de espacios vectoriales"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dado un espacio vectorial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9123abddc2ec35f72035ec59f443c79ee052c9ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.163ex; height:2.176ex;" alt="{\displaystyle E\,}"></span> y un subespacio vectorial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\subset E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>⊂</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\subset E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dc790c91db34007686074e392dfed5bb12f482e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.615ex; height:2.176ex;" alt="{\displaystyle F\subset E}"></span>. </p><p>Dados <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v\in E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>∈</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u,v\in E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f83411290ec9b03db58a85bf1ddbfb428bf00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.108ex; height:2.509ex;" alt="{\displaystyle u,v\in E}"></span> diremos que están <a href="/wiki/Relaci%C3%B3n_binaria" title="Relación binaria">relacionados</a> módulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af686cd9f8a742bde6d8ee773ddebc793960d0ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.128ex; height:2.176ex;" alt="{\displaystyle F\,}"></span> si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u-v\in F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>−</mo> <mi>v</mi> <mo>∈</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u-v\in F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eba9220fe624bfb5f327e39f995c54deefedaf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.879ex; height:2.343ex;" alt="{\displaystyle u-v\in F}"></span>. </p> <ul><li>La relación anterior es una <a href="/wiki/Relaci%C3%B3n_de_equivalencia" title="Relación de equivalencia">relación de equivalencia</a>.</li></ul> <dl><dd>Se nota por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [u]=u+F:=\{u+v:v\in F\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>u</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>u</mi> <mo>+</mo> <mi>F</mi> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo>:</mo> <mi>v</mi> <mo>∈</mo> <mi>F</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [u]=u+F:=\{u+v:v\in F\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa427a3d2f6e6dff1e7a4311d5acee291f2ec173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.647ex; height:2.843ex;" alt="{\displaystyle [u]=u+F:=\{u+v:v\in F\}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\{w:w=u+v,\;v\in F\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>w</mi> <mo>:</mo> <mi>w</mi> <mo>=</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>v</mi> <mo>∈</mo> <mi>F</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\{w:w=u+v,\;v\in F\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/831bff6b2b1ebeab022f9a45d0cde0ebd1b488c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.828ex; height:2.843ex;" alt="{\displaystyle =\{w:w=u+v,\;v\in F\}}"></span> a la clase de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/880f91e25cd451d89d1f6d0d06852b56a7b74a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle u\,}"></span> módulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af686cd9f8a742bde6d8ee773ddebc793960d0ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.128ex; height:2.176ex;" alt="{\displaystyle F\,}"></span>.</dd></dl> <p>Llamaremos <b><a href="/wiki/Espacio_cociente_(%C3%A1lgebra_lineal)" title="Espacio cociente (álgebra lineal)">conjunto cociente</a></b> o <b>espacio cociente</b> al conjunto de las clases de equivalencia anterior: </p> <dl><dd>Se nota por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E/F_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E/F_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df2c8bfaaf52d537410fdb4bb019f6761f8683cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.679ex; height:2.843ex;" alt="{\displaystyle E/F_{}^{}}"></span> a dicho espacio cociente.</dd></dl> <p>El espacio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E/F_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E/F_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df2c8bfaaf52d537410fdb4bb019f6761f8683cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.679ex; height:2.843ex;" alt="{\displaystyle E/F_{}^{}}"></span> es un espacio vectorial con las operaciones siguientes: </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}[u]+[v]&:=&[u+v]\\\;\;\;\;\;\;\;\lambda [u]&:=&[\lambda u]\;\;\;\;\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">[</mo> <mi>u</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>v</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mo>:=</mo> </mtd> <mtd> <mo stretchy="false">[</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>λ</mi> <mo stretchy="false">[</mo> <mi>u</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mo>:=</mo> </mtd> <mtd> <mo stretchy="false">[</mo> <mi>λ</mi> <mi>u</mi> <mo stretchy="false">]</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}[u]+[v]&:=&[u+v]\\\;\;\;\;\;\;\;\lambda [u]&:=&[\lambda u]\;\;\;\;\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca3cd7076e7a8c1cb3fb4e10e740d11756ecb47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.938ex; height:6.176ex;" alt="{\displaystyle {\begin{matrix}[u]+[v]&:=&[u+v]\\\;\;\;\;\;\;\;\lambda [u]&:=&[\lambda u]\;\;\;\;\end{matrix}}}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Construcciones_básicas"><span id="Construcciones_b.C3.A1sicas"></span>Construcciones básicas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=34" title="Editar sección: Construcciones básicas"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Además de lo expuesto en los ejemplos anteriores, hay una serie de construcciones que nos proporcionan espacios vectoriales a partir de otros. Además de las definiciones concretas que figuran a continuación, también se caracterizan por <a href="/w/index.php?title=Propiedades_universales&action=edit&redlink=1" class="new" title="Propiedades universales (aún no redactado)">propiedades universales</a>, que determina un objeto <i>X</i> especificando las aplicaciones lineales de <i>X</i> a cualquier otro espacio vectorial. </p> <div class="mw-heading mw-heading3"><h3 id="Suma_directa_de_espacios_vectoriales">Suma directa de espacios vectoriales</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=35" title="Editar sección: Suma directa de espacios vectoriales"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dado dos espacios vectoriales <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E,\;F_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>,</mo> <mspace width="thickmathspace" /> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E,\;F_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c0030578ea4f9eaeb0907912387cdbd045de9b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.196ex; height:2.509ex;" alt="{\displaystyle E,\;F_{}^{}}"></span> sobre un mismo cuerpo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a6c0f78592eaf8c5b477d9551ea3adffeea33fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K_{}^{}}"></span>, llamaremos suma directa al espacio vectorial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\times F=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>×</mo> <mi>F</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\times F=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bbd72bdcd1232b18ab8812e638dcffba189d37e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.81ex; height:2.176ex;" alt="{\displaystyle E\times F=}"></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{u:=(u_{1},\;u_{2}):u_{1}\in E,\;u_{2}\in F\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>u</mi> <mo>:=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mi>E</mi> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mi>F</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{u:=(u_{1},\;u_{2}):u_{1}\in E,\;u_{2}\in F\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33789a7480c77fb654f4a88bbf09060c9decf402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.238ex; height:2.843ex;" alt="{\displaystyle \{u:=(u_{1},\;u_{2}):u_{1}\in E,\;u_{2}\in F\}}"></span>, veamos que están bien definidas las dos operaciones: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u+v=(u_{1},\;u_{2})+(v_{1},\;v_{2})=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u+v=(u_{1},\;u_{2})+(v_{1},\;v_{2})=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8336b2580b4ed48bc846a00787659c559f243fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.798ex; height:2.843ex;" alt="{\displaystyle u+v=(u_{1},\;u_{2})+(v_{1},\;v_{2})=}"></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u_{1}+v_{1},\;u_{2}+v_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u_{1}+v_{1},\;u_{2}+v_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d286da2688cb34b8a47217b0d6b31535bc25e8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.301ex; height:2.843ex;" alt="{\displaystyle (u_{1}+v_{1},\;u_{2}+v_{2})}"></span>,</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle au=a(u_{1},\;u_{2})=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>u</mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle au=a(u_{1},\;u_{2})=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05da200e02af8f677275401d639c2d7c0c560ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.597ex; height:2.843ex;" alt="{\displaystyle au=a(u_{1},\;u_{2})=}"></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (au_{1},\;au_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mi>a</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (au_{1},\;au_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/924a56ad5dbbb3806a5397ed133a41d8d4a3b391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.716ex; height:2.843ex;" alt="{\displaystyle (au_{1},\;au_{2})}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Espacios_vectoriales_con_estructura_adicional">Espacios vectoriales con estructura adicional</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=36" title="Editar sección: Espacios vectoriales con estructura adicional"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Desde el punto de vista del álgebra lineal, los espacios vectoriales se comprenden completamente en la medida en que cualquier espacio vectorial se caracteriza, salvo isomorfismos, por su dimensión. Sin embargo, los espacios vectoriales <i>ad hoc</i> no ofrecen un marco para hacer frente a la cuestión fundamental para el análisis de si una sucesión de funciones <a href="/wiki/L%C3%ADmite_de_una_sucesi%C3%B3n" title="Límite de una sucesión">converge</a> a otra función. Asimismo, el álgebra lineal no está adaptada <i>per se</i> para hacer frente a <a href="/wiki/Series_infinitas" class="mw-redirect" title="Series infinitas">series infinitas</a>, ya que la suma solo permite un número finito de términos para sumar. <cite id="labelFunctionalAnalysis">Las necesidades del <a href="/wiki/An%C3%A1lisis_funcional" title="Análisis funcional">análisis funcional</a> requieren considerar nuevas estructuras.</cite> </p> <div class="mw-heading mw-heading3"><h3 id="Espacios_normados">Espacios normados</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=37" title="Editar sección: Espacios normados"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint AP rellink"><span style="font-size:88%">Artículos principales:</span> <i><a href="/wiki/Espacio_vectorial_normado" title="Espacio vectorial normado"> Espacio vectorial normado</a></i><span style="font-size:88%"> y </span><i><a href="/wiki/Norma_(matem%C3%A1ticas)" class="mw-redirect" title="Norma (matemáticas)"> Norma (matemáticas)</a></i>.</div> <p>Un espacio vectorial es <b><a href="/wiki/Espacio_vectorial_normado" title="Espacio vectorial normado">normado</a></b> si está dotado de una <a href="/wiki/Norma_(matem%C3%A1ticas)" class="mw-redirect" title="Norma (matemáticas)">norma</a>. </p> <dl><dd><b>Proposición 5</b>. Un espacio normado es un <a href="/wiki/Espacio_m%C3%A9trico" title="Espacio métrico">espacio métrico</a>, donde la distancia viene dada por:</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=\|x-y\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=\|x-y\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94df456fedac6b08c33cb4dffa5345a0ce0891f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.293ex; height:2.843ex;" alt="{\displaystyle d(x,y)=\|x-y\|}"></span></dd></dl> <p>Toda distancia inducida por la norma es una distancia. </p> <div class="mw-heading mw-heading3"><h3 id="Espacios_vectoriales_topológicos"><span id="Espacios_vectoriales_topol.C3.B3gicos"></span>Espacios vectoriales topológicos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=38" title="Editar sección: Espacios vectoriales topológicos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint AP rellink"><span style="font-size:88%">Artículo principal:</span> <i><a href="/wiki/Espacio_vectorial_topol%C3%B3gico" title="Espacio vectorial topológico"> Espacio vectorial topológico</a></i></div> <p>Dada una topología <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16885dadcfd11c9155dab24ef46c98f8473dd344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau _{}^{}}"></span> sobre un espacio vectorial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dd890de269ef36ebe48b74d7515c9ae8081caa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X_{}^{}}"></span> donde los puntos sean cerrados y las dos operaciones del espacio vectorial sean continuas respecto dichas topología, diremos que: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16885dadcfd11c9155dab24ef46c98f8473dd344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau _{}^{}}"></span> es una <b>topología vectorial</b> sobre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dd890de269ef36ebe48b74d7515c9ae8081caa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X_{}^{}}"></span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dd890de269ef36ebe48b74d7515c9ae8081caa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X_{}^{}}"></span> es un <b>espacio vectorial topológico</b>.</li></ul> <dl><dd><b>Proposición 6.</b>. Todo espacio vectorial topológico dotado de una métrica es espacio normado.</dd> <dd><b>Proposición 7.</b>. Todo espacio normado es un espacio vectorial topológico.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Espacios_de_Banach">Espacios de Banach</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=39" title="Editar sección: Espacios de Banach"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint AP rellink"><span style="font-size:88%">Artículo principal:</span> <i><a href="/wiki/Espacio_de_Banach" title="Espacio de Banach"> Espacio de Banach</a></i></div> <p>Un espacio de <b>Banach</b> es un espacio normado y completo. </p> <div class="mw-heading mw-heading3"><h3 id="Espacios_prehilbertianos">Espacios prehilbertianos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=40" title="Editar sección: Espacios prehilbertianos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint AP rellink"><span style="font-size:88%">Artículo principal:</span> <i><a href="/wiki/Espacio_prehilbertiano" title="Espacio prehilbertiano"> Espacio prehilbertiano</a></i></div> <p>Un <b><a href="/wiki/Espacio_prehilbertiano" title="Espacio prehilbertiano">espacio prehilbertiano</a></b> es un par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E_{}^{},\langle \cdot |\cdot \rangle )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo>,</mo> <mo fence="false" stretchy="false">⟨</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E_{}^{},\langle \cdot |\cdot \rangle )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c31cb91c9b51e604be52ca1d70791edc565f732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.369ex; height:2.843ex;" alt="{\displaystyle (E_{}^{},\langle \cdot |\cdot \rangle )}"></span>, donde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28f93f365b478d2995b9c06f677212f50382aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E_{}^{}}"></span> es un espacio vectorial y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot |\cdot \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \cdot |\cdot \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d36ddc87338be38721a2d721d1a6cc9ab294474a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.75ex; height:2.843ex;" alt="{\displaystyle \langle \cdot |\cdot \rangle }"></span> es un <a href="/wiki/Producto_escalar" title="Producto escalar">producto a escalar</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Espacios_de_Hilbert">Espacios de Hilbert</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=41" title="Editar sección: Espacios de Hilbert"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint AP rellink"><span style="font-size:88%">Artículo principal:</span> <i><a href="/wiki/Espacio_de_Hilbert" title="Espacio de Hilbert"> Espacio de Hilbert</a></i></div> <p>Un <b>espacio de Hilbert</b> es un espacio prehilbertiano completo por la norma definida por el producto escalar. </p> <div class="mw-heading mw-heading2"><h2 id="Morfismos_entre_espacios_vectoriales">Morfismos entre espacios vectoriales</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=42" title="Editar sección: Morfismos entre espacios vectoriales"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Son <a href="/wiki/Teor%C3%ADa_de_categor%C3%ADas" title="Teoría de categorías">aplicaciones entre espacios vectoriales</a> que mantienen la estructura de los espacios vectoriales, es decir, conservan las dos operaciones y las propiedades de estas de uno a otro de dichos espacios. </p> <div class="mw-heading mw-heading3"><h3 id="Aplicaciones_lineales">Aplicaciones lineales</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=43" title="Editar sección: Aplicaciones lineales"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint AP rellink"><span style="font-size:88%">Artículo principal:</span> <i><a href="/wiki/Aplicaci%C3%B3n_lineal" title="Aplicación lineal"> Aplicación lineal</a></i></div> <p>Dado dos espacios vectoriales <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28f93f365b478d2995b9c06f677212f50382aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E_{}^{}}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba69787528be36f0312673a5ffe86596b55f4a24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F_{}^{}}"></span>, sobre un mismo cuerpo, diremos que una aplicación <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:E\rightarrow F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:E\rightarrow F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd8d46b895208da4ebe2c3df74db65cf3f1e963" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.346ex; height:2.509ex;" alt="{\displaystyle f:E\rightarrow F}"></span> es <b>lineal</b> si: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(u+_{E}v)=f(u)+_{F}f(v)_{}^{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(u+_{E}v)=f(u)+_{F}f(v)_{}^{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b51a6fd8165311d4cc250f8690b2cebc89836a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.908ex; height:2.843ex;" alt="{\displaystyle f(u+_{E}v)=f(u)+_{F}f(v)_{}^{}}"></span>,</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a\cdot _{E}u)=a\cdot _{F}f(u)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(a\cdot _{E}u)=a\cdot _{F}f(u)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d36d20fd102f8e5e87a154ec9ee7db3fcb2289b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.702ex; height:2.843ex;" alt="{\displaystyle f(a\cdot _{E}u)=a\cdot _{F}f(u)}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Véase_también"><span id="V.C3.A9ase_tambi.C3.A9n"></span>Véase también</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=44" title="Editar sección: Véase también"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Combinaci%C3%B3n_lineal" title="Combinación lineal">Combinación lineal</a></li> <li><a href="/wiki/Sistema_generador" title="Sistema generador">Sistema generador</a></li> <li><a href="/wiki/Independencia_lineal" class="mw-redirect" title="Independencia lineal">Independencia lineal</a></li> <li><a href="/wiki/Base_(%C3%A1lgebra)" title="Base (álgebra)">Base (álgebra)</a></li> <li><a href="/wiki/Teorema_rango-nulidad" title="Teorema rango-nulidad">Teorema rango-nulidad</a></li> <li><a href="/wiki/Ortogonal" class="mw-redirect" title="Ortogonal">Base Ortogonal</a></li> <li><a href="/wiki/Ortonormal" title="Ortonormal">Base Ortonormal</a></li> <li><a href="/wiki/Coordenadas_cartesianas" title="Coordenadas cartesianas">Coordenadas cartesianas</a></li> <li><a href="/wiki/Producto_escalar" title="Producto escalar">Producto escalar</a></li> <li><a href="/wiki/Producto_vectorial" title="Producto vectorial">Producto vectorial</a></li> <li><a href="/wiki/Producto_mixto" title="Producto mixto">Producto mixto</a></li> <li><a href="/wiki/Producto_tensorial" title="Producto tensorial">Producto tensorial</a></li></ul> <ul><li><span typeof="mw:File"><a href="/wiki/Archivo:Nuvola_apps_edu_mathematics-p.svg" class="mw-file-description" title="Ver el portal sobre Matemática"><img alt="Ver el portal sobre Matemática" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/20px-Nuvola_apps_edu_mathematics-p.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/30px-Nuvola_apps_edu_mathematics-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/40px-Nuvola_apps_edu_mathematics-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> <a href="/wiki/Portal:Matem%C3%A1tica" title="Portal:Matemática">Portal:Matemática</a>. Contenido relacionado con <b><a href="/wiki/Matem%C3%A1tica" class="mw-redirect" title="Matemática">Matemática</a></b>.</li> <li><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/15px-Wikibooks-logo.svg.png" decoding="async" width="15" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/23px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/30px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></span></span> <a href="/wiki/Wikilibros" title="Wikilibros">Wikilibros</a> alberga un libro o manual sobre <b><a href="https://es.wikibooks.org/wiki/%C3%81lgebra" class="extiw" title="b:Álgebra">Espacio vectorial</a></b>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Referencias">Referencias</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=45" title="Editar sección: Referencias"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="listaref" style="-moz-column-count:2; -webkit-column-count:2; column-count:2; list-style-type: decimal;"></div> <div class="mw-heading mw-heading3"><h3 id="Notas">Notas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=46" title="Editar sección: Notas"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a href="#CITAREFBourbaki1969">Bourbaki, 1969</a>, ch. "Álgabre linéaire et álgebre multilinéaire", pp. 78–91.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a href="#CITAREFBolzano1804">Bolzano, 1804</a>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><a href="#CITAREFMöbius1827">Möbius, 1827</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><a href="#CITAREFHamilton1853">Hamilton, 1853</a>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text"><a href="#CITAREFGrassmann1844">Grassmann, 1844</a>.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text"><a href="#CITAREFPeano1888">Peano, 1888</a>, ch. IX.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><a href="#CITAREFBanach1922">Banach, 1922</a>.</span> </li> </ol> <div class="mw-heading mw-heading3"><h3 id="Referencias_históricas"><span id="Referencias_hist.C3.B3ricas"></span>Referencias históricas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=47" title="Editar sección: Referencias históricas"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span id="CITAREFBanach1922" class="citation libro"><a href="/wiki/Stefan_Banach" title="Stefan Banach">Banach, Stefan</a> (1922). <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3120.pdf"><i>Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales (On operations in abstract sets and their application to integral equations)</i></a> <span style="color:var(--color-subtle, #555 );">(en francés)</span> <b>3</b>. <a href="/wiki/Fundamenta_Mathematicae" title="Fundamenta Mathematicae">Fundamenta Mathematicae</a>. <small><span class="plainlinks"><a href="/wiki/ISSN" class="mw-redirect" title="ISSN">ISSN</a> <a rel="nofollow" class="external text" href="//portal.issn.org/resource/ISSN/0016-2736">0016-2736</a></span></small>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEspacio+vectorial&rft.au=Banach%2C+Stefan&rft.aufirst=Stefan&rft.aulast=Banach&rft.btitle=Sur+les+op%C3%A9rations+dans+les+ensembles+abstraits+et+leur+application+aux+%C3%A9quations+int%C3%A9grales+%28On+operations+in+abstract+sets+and+their+application+to+integral+equations%29&rft.date=1922&rft.genre=book&rft.pub=Fundamenta+Mathematicae&rft.volume=3&rft_id=http%3A%2F%2Fmatwbn.icm.edu.pl%2Fksiazki%2Ffm%2Ffm3%2Ffm3120.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFBolzano1804" class="citation libro"><a href="/wiki/Bernard_Bolzano" title="Bernard Bolzano">Bolzano, Bernard</a> (1804). <a rel="nofollow" class="external text" href="http://dml.cz/handle/10338.dmlcz/400338"><i>Betrachtungen über einige Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry)</i></a> <span style="color:var(--color-subtle, #555 );">(en alemán)</span>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEspacio+vectorial&rft.au=Bolzano%2C+Bernard&rft.aufirst=Bernard&rft.aulast=Bolzano&rft.btitle=Betrachtungen+%C3%BCber+einige+Gegenst%C3%A4nde+der+Elementargeometrie+%28Considerations+of+some+aspects+of+elementary+geometry%29&rft.date=1804&rft.genre=book&rft_id=http%3A%2F%2Fdml.cz%2Fhandle%2F10338.dmlcz%2F400338&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFBourbaki1969" class="citation libro"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (1969). <i>Éléments d'histoire des mathématiques (Elements of history of mathematics)</i> <span style="color:var(--color-subtle, #555 );">(en francés)</span>. París: Hermann.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEspacio+vectorial&rft.au=Bourbaki%2C+Nicolas&rft.aufirst=Nicolas&rft.aulast=Bourbaki&rft.btitle=%C3%89l%C3%A9ments+d%27histoire+des+math%C3%A9matiques+%28Elements+of+history+of+mathematics%29&rft.date=1969&rft.genre=book&rft.place=Par%C3%ADs&rft.pub=Hermann&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFGrassmann1844" class="citation libro"><a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Grassmann, Hermann</a> (1844). <a rel="nofollow" class="external text" href="http://books.google.com/books?id=bKgAAAAAMAAJ&pg=PA1&dq=Die+Lineale+Ausdehnungslehre+ein+neuer+Zweig+der+Mathematik"><i>Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik</i></a> <span style="color:var(--color-subtle, #555 );">(en alemán)</span>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEspacio+vectorial&rft.au=Grassmann%2C+Hermann&rft.aufirst=Hermann&rft.aulast=Grassmann&rft.btitle=Die+Lineale+Ausdehnungslehre+-+Ein+neuer+Zweig+der+Mathematik&rft.date=1844&rft.genre=book&rft_id=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbKgAAAAAMAAJ%26pg%3DPA1%26dq%3DDie%2BLineale%2BAusdehnungslehre%2Bein%2Bneuer%2BZweig%2Bder%2BMathematik&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFHamilton1853" class="citation libro"><a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton, William Rowan</a> (1853). <a rel="nofollow" class="external text" href="http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05230001&seq=9"><i>Lectures on Quaternions</i></a> <span style="color:var(--color-subtle, #555 );">(en inglés)</span>. Royal Irish Academy.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEspacio+vectorial&rft.au=Hamilton%2C+William+Rowan&rft.aufirst=William+Rowan&rft.aulast=Hamilton&rft.btitle=Lectures+on+Quaternions&rft.date=1853&rft.genre=book&rft.pub=Royal+Irish+Academy&rft_id=http%3A%2F%2Fhistorical.library.cornell.edu%2Fcgi-bin%2Fcul.math%2Fdocviewer%3Fdid%3D05230001%26seq%3D9&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFMöbius1827" class="citation libro"><a href="/wiki/August_Ferdinand_M%C3%B6bius" class="mw-redirect" title="August Ferdinand Möbius">Möbius, August Ferdinand</a> (1827). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090412013616/http://mathdoc.emath.fr/cgi-bin/oeitem?id=OE_MOBIUS__1_1_0"><i>Der Barycentrische Calcul : ein neues Hülfsmittel zur analytischen Behandlung der Geometrie (Barycentric calculus: a new utility for an analytic treatment of geometry)</i></a> <span style="color:var(--color-subtle, #555 );">(en alemán)</span>. Archivado desde <a rel="nofollow" class="external text" href="http://mathdoc.emath.fr/cgi-bin/oeitem?id=OE_MOBIUS__1_1_0">el original</a> el 12 de abril de 2009.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEspacio+vectorial&rft.au=M%C3%B6bius%2C+August+Ferdinand&rft.aufirst=August+Ferdinand&rft.aulast=M%C3%B6bius&rft.btitle=Der+Barycentrische+Calcul+%3A+ein+neues+H%C3%BClfsmittel+zur+analytischen+Behandlung+der+Geometrie+%28Barycentric+calculus%3A+a+new+utility+for+an+analytic+treatment+of+geometry%29&rft.date=1827&rft.genre=book&rft_id=http%3A%2F%2Fmathdoc.emath.fr%2Fcgi-bin%2Foeitem%3Fid%3DOE_MOBIUS__1_1_0&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFMoore1995" class="citation">Moore, Gregory H. (1995), «The axiomatization of linear algebra: 1875–1940», <i><a href="/wiki/Historia_Mathematica" title="Historia Mathematica">Historia Mathematica</a></i> <b>22</b> (3): 262-303, <small><a href="/wiki/ISSN" class="mw-redirect" title="ISSN">ISSN</a> <a rel="nofollow" class="external text" href="//portal.issn.org/resource/issn/0315-0860">0315-0860</a></small></span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEspacio+vectorial&rft.atitle=The+axiomatization+of+linear+algebra%3A+1875%E2%80%931940&rft.au=Moore%2C+Gregory+H.&rft.aufirst=Gregory+H.&rft.aulast=Moore&rft.date=1995&rft.genre=article&rft.issn=0315-0860&rft.issue=3&rft.jtitle=Historia+Mathematica&rft.pages=262-303&rft.volume=22&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><span id="CITAREFPeano1888" class="citation libro"><a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Peano, Giuseppe</a> (1888). <i>Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva</i> <span style="color:var(--color-subtle, #555 );">(en italiano)</span>. Turin.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEspacio+vectorial&rft.au=Peano%2C+Giuseppe&rft.aufirst=Giuseppe&rft.aulast=Peano&rft.btitle=Calcolo+Geometrico+secondo+l%27Ausdehnungslehre+di+H.+Grassmann+preceduto+dalle+Operazioni+della+Logica+Deduttiva&rft.date=1888&rft.genre=book&rft.place=Turin&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Bibliografía"><span id="Bibliograf.C3.ADa"></span>Bibliografía</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=48" title="Editar sección: Bibliografía"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span id="CITAREFCastellet,_M.Llerena,_I.1988" class="citation libro">Castellet, M.; Llerena, I. (1988). «IV espais vectorials». <i>Àlgebra lineal i geometría</i> <span style="color:var(--color-subtle, #555 );">(en catalán)</span>. Publ. UAB.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEspacio+vectorial&rft.atitle=%C3%80lgebra+lineal+i+geometr%C3%ADa&rft.au=Castellet%2C+M.&rft.au=Llerena%2C+I.&rft.aulast=Castellet%2C+M.&rft.btitle=IV+espais+vectorials&rft.date=1988&rft.genre=bookitem&rft.pub=Publ.+UAB&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFLang,_S.1976" class="citation libro">Lang, S. (1976). <i>Álgebra Lineal</i>. Fondo Educativo Interamericano.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEspacio+vectorial&rft.au=Lang%2C+S.&rft.aulast=Lang%2C+S.&rft.btitle=%C3%81lgebra+Lineal&rft.date=1976&rft.genre=book&rft.pub=Fondo+Educativo+Interamericano&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment"><code>|fechaacceso=</code> requiere <code>|url=</code> (<a href="/wiki/Ayuda:Errores_en_las_referencias#accessdate_missing_url" title="Ayuda:Errores en las referencias">ayuda</a>)</span></li> <li>Queysanne, M., <i>Álgebra Básica</i>, Vicens-Vives. 1973.</li> <li>Rudin, w., <i>Análisis Funcional</i> (Definición axiomática de espacios vectoriales topológicos introductivamente), Reverté.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Enlaces_externos">Enlaces externos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espacio_vectorial&action=edit&section=49" title="Editar sección: Enlaces externos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20051001062338/http://www.frontiernet.net/~imaging/vector_calculator.html">Juega con vectores</a></li> <li><span id="Reference-Mathworld-Espacio_vectorial" class="citation web"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W</a>. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/VectorSpace.html">«Espacio vectorial»</a>. En Weisstein, Eric W, ed. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i> <span style="color:var(--color-subtle, #555 );">(en inglés)</span>. <a href="/wiki/Wolfram_Research" title="Wolfram Research">Wolfram Research</a>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEspacio+vectorial&rft.atitle=Espacio+vectorial&rft.au=Weisstein%2C+Eric+W&rft.aulast=Weisstein%2C+Eric+W&rft.genre=article&rft.jtitle=MathWorld&rft.pub=Wolfram+Research&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FVectorSpace.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></li> <li><a rel="nofollow" class="external text" href="http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-9-independence-basis-and-dimension/">A lecture</a> about fundamental concepts related to vector spaces (given at <a href="/wiki/MIT" class="mw-redirect" title="MIT">MIT</a>)</li> <li><a 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