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vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ortogonalidad_en_espacios_vectoriales"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Ortogonalidad en espacios vectoriales</span> </div> </a> <button aria-controls="toc-Ortogonalidad_en_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 Ortogonalidad en espacios vectoriales</span> </button> <ul id="toc-Ortogonalidad_en_espacios_vectoriales-sublist" class="vector-toc-list"> <li id="toc-Definición" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definición"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Definición</span> </div> </a> <ul id="toc-Definición-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complemento_ortogonal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complemento_ortogonal"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Complemento ortogonal</span> </div> </a> <ul id="toc-Complemento_ortogonal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ortogonalidad_y_perpendicularidad" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ortogonalidad_y_perpendicularidad"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Ortogonalidad y perpendicularidad</span> </div> </a> <ul id="toc-Ortogonalidad_y_perpendicularidad-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ortogonalidad_respecto_de_una_matriz_(A-ortogonalidad)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ortogonalidad_respecto_de_una_matriz_(A-ortogonalidad)"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Ortogonalidad respecto de una matriz (A-ortogonalidad)</span> </div> </a> <ul id="toc-Ortogonalidad_respecto_de_una_matriz_(A-ortogonalidad)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Transformación_ortogonal" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Transformación_ortogonal"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Transformación ortogonal</span> </div> </a> <ul id="toc-Transformación_ortogonal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ortogonalidad_en_otros_contextos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ortogonalidad_en_otros_contextos"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Ortogonalidad en otros contextos</span> </div> </a> <ul id="toc-Ortogonalidad_en_otros_contextos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sistemas_de_coordenadas_ortogonales" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sistemas_de_coordenadas_ortogonales"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Sistemas de coordenadas ortogonales</span> </div> </a> <ul id="toc-Sistemas_de_coordenadas_ortogonales-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Véase_también" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Véase_también"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referencias"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Referencias</span> </div> </a> <ul id="toc-Referencias-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">Ortogonalidad (matemática)</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 38 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-38" 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">38 idiomas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%B9%D8%A7%D9%85%D8%AF_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="تعامد (رياضيات) (árabe)" lang="ar" hreflang="ar" data-title="تعامد (رياضيات)" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Ortoqonal" title="Ortoqonal (azerbaiyano)" lang="az" hreflang="az" data-title="Ortoqonal" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaiyano" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D1%80%D1%82%D0%BE%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D0%BD%D0%BE%D1%81%D1%82" title="Ортогоналност (búlgaro)" lang="bg" hreflang="bg" data-title="Ортогоналност" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Ortogonal" title="Ortogonal (catalán)" lang="ca" hreflang="ca" data-title="Ortogonal" data-language-autonym="Català" data-language-local-name="catalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Ortogonalita" title="Ortogonalita (checo)" lang="cs" hreflang="cs" data-title="Ortogonalita" 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%9E%D1%80%D1%82%D0%BE%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D0%BB%C4%95%D1%85" title="Ортогоналлĕх (chuvasio)" lang="cv" hreflang="cv" data-title="Ортогоналлĕх" data-language-autonym="Чӑвашла" data-language-local-name="chuvasio" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Ortogonalitet" title="Ortogonalitet (danés)" lang="da" hreflang="da" data-title="Ortogonalitet" 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/Orthogonalit%C3%A4t" title="Orthogonalität (alemán)" lang="de" hreflang="de" data-title="Orthogonalität" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Orthogonality" title="Orthogonality (inglés)" lang="en" hreflang="en" data-title="Orthogonality" 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/Orteco" title="Orteco (esperanto)" lang="eo" hreflang="eo" data-title="Orteco" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Ortogonal" title="Ortogonal (euskera)" lang="eu" hreflang="eu" data-title="Ortogonal" 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/%D8%AA%D8%B9%D8%A7%D9%85%D8%AF_(%D8%AC%D8%A8%D8%B1_%D8%AE%D8%B7%DB%8C)" title="تعامد (جبر خطی) (persa)" lang="fa" hreflang="fa" data-title="تعامد (جبر خطی)" 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/Kohtisuoruus" title="Kohtisuoruus (finés)" lang="fi" hreflang="fi" data-title="Kohtisuoruus" 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/Orthogonalit%C3%A9" title="Orthogonalité (francés)" lang="fr" hreflang="fr" data-title="Orthogonalité" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%95%D7%A8%D7%AA%D7%95%D7%92%D7%95%D7%A0%D7%9C%D7%99%D7%95%D7%AA" title="אורתוגונליות (hebreo)" lang="he" hreflang="he" data-title="אורתוגונליות" data-language-autonym="עברית" data-language-local-name="hebreo" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Mer%C5%91legess%C3%A9g" title="Merőlegesség (húngaro)" lang="hu" hreflang="hu" data-title="Merőlegesség" 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%88%D6%82%D5%B2%D5%B2%D5%A1%D5%B6%D5%AF%D5%B5%D5%B8%D6%82%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Ուղղանկյունություն (armenio)" lang="hy" hreflang="hy" data-title="Ուղղանկյունություն" data-language-autonym="Հայերեն" data-language-local-name="armenio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%9B%B4%E4%BA%A4" title="直交 (japonés)" lang="ja" hreflang="ja" data-title="直交" 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/%EC%A7%81%EA%B5%90" title="직교 (coreano)" lang="ko" hreflang="ko" data-title="직교" 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%9E%D1%80%D1%82%D0%BE%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D0%B4%D1%83%D1%83%D0%BB%D1%83%D0%BA" title="Ортогоналдуулук (kirguís)" lang="ky" hreflang="ky" data-title="Ортогоналдуулук" data-language-autonym="Кыргызча" data-language-local-name="kirguís" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9E%D1%80%D1%82%D0%BE%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D0%BD%D0%BE%D1%81%D1%82" title="Ортогоналност (macedonio)" lang="mk" hreflang="mk" data-title="Ортогоналност" data-language-autonym="Македонски" data-language-local-name="macedonio" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Orthogonaal" title="Orthogonaal (neerlandés)" lang="nl" hreflang="nl" data-title="Orthogonaal" 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/Ortogonalitet" title="Ortogonalitet (noruego nynorsk)" lang="nn" hreflang="nn" data-title="Ortogonalitet" 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/Ortogonalitet" title="Ortogonalitet (noruego bokmal)" lang="nb" hreflang="nb" data-title="Ortogonalitet" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ortogonalno%C5%9B%C4%87" title="Ortogonalność (polaco)" lang="pl" hreflang="pl" data-title="Ortogonalność" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Ortogonalidade" title="Ortogonalidade (portugués)" lang="pt" hreflang="pt" data-title="Ortogonalidade" data-language-autonym="Português" data-language-local-name="portugués" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Ortogonalitate" title="Ortogonalitate (rumano)" lang="ro" hreflang="ro" data-title="Ortogonalitate" 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%9E%D1%80%D1%82%D0%BE%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Ортогональность (ruso)" lang="ru" hreflang="ru" data-title="Ортогональность" data-language-autonym="Русский" data-language-local-name="ruso" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Ortogonalnost" title="Ortogonalnost (serbocroata)" lang="sh" hreflang="sh" data-title="Ortogonalnost" 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-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Ortogonalnost" title="Ortogonalnost (esloveno)" lang="sl" hreflang="sl" data-title="Ortogonalnost" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9E%D1%80%D1%82%D0%BE%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D0%BD%D0%BE%D1%81%D1%82" title="Ортогоналност (serbio)" lang="sr" hreflang="sr" data-title="Ортогоналност" 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/Ortogonalitet" title="Ortogonalitet (sueco)" lang="sv" hreflang="sv" data-title="Ortogonalitet" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Ortogonalidad" title="Ortogonalidad (tagalo)" lang="tl" hreflang="tl" data-title="Ortogonalidad" 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/Ortogonallik" title="Ortogonallik (turco)" lang="tr" hreflang="tr" data-title="Ortogonallik" 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%9E%D1%80%D1%82%D0%BE%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96%D1%81%D1%82%D1%8C" title="Ортогональність (ucraniano)" lang="uk" hreflang="uk" data-title="Ортогональність" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li 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href="/w/index.php?title=Ortogonal&redirect=no" class="mw-redirect" title="Ortogonal">Ortogonal</a>»)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="es" dir="ltr"><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Archivo:3D_coordinate_system.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/3D_coordinate_system.svg/350px-3D_coordinate_system.svg.png" decoding="async" width="350" height="350" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/3D_coordinate_system.svg/525px-3D_coordinate_system.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/3D_coordinate_system.svg/700px-3D_coordinate_system.svg.png 2x" data-file-width="487" data-file-height="487" /></a><figcaption>Tres <a href="/wiki/Plano_(geometr%C3%ADa)" title="Plano (geometría)">planos</a> ortogonales</figcaption></figure> <p>En <a href="/wiki/Matem%C3%A1ticas" title="Matemáticas">matemáticas</a>, el término <b>ortogonalidad</b> (del griego <i>ὀρθός</i> ‘recto’ y <i>γωνία</i> ‘ángulo’) es una generalización de la noción <a href="/wiki/Geometr%C3%ADa" title="Geometría">geométrica</a> de <a href="/wiki/Perpendicularidad" title="Perpendicularidad">perpendicularidad</a>. En el <a href="/wiki/Espacio_eucl%C3%ADdeo" title="Espacio euclídeo">espacio euclídeo</a> convencional, el término ortogonal y el término perpendicular son sinónimos. Sin embargo, en espacios de dimensión finita y en <a href="/wiki/Geometr%C3%ADas_no_eucl%C3%ADdeas" class="mw-redirect" title="Geometrías no euclídeas">geometrías no euclídeas</a>, el concepto de ortogonalidad generaliza al de perpendicularidad. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Ortogonalidad_en_espacios_vectoriales">Ortogonalidad en espacios vectoriales</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ortogonalidad_(matem%C3%A1tica)&action=edit&section=1" title="Editar sección: Ortogonalidad en espacios vectoriales"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Definición"><span id="Definici.C3.B3n"></span>Definición</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ortogonalidad_(matem%C3%A1tica)&action=edit&section=2" title="Editar sección: Definición"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Formalmente, en un <a href="/wiki/Espacio_de_producto_interior" class="mw-redirect" title="Espacio de producto interior">espacio vectorial con producto interior</a> <i>V</i>, dos <a href="/wiki/Vector_(matem%C3%A1tica)" class="mw-redirect" title="Vector (matemática)">vectores</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 x\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa374e20b2db7f6b8caa71ff1865f7f84f215c9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.958ex; height:2.176ex;" alt="{\displaystyle x\in V}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da6483ea92075167a597963b8d1cb1f398d173f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle y\in V}"></span> son ortogonales si el <a href="/wiki/Producto_escalar" title="Producto escalar">producto escalar</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 \langle x,y\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 \langle x,y\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df1806ebe1fed1a728b18aed82c30be8b2a0acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle }"></span> es cero.<sup id="cite_ref-1" class="reference separada"><a href="#cite_note-1"><span class="corchete-llamada">[</span>1<span class="corchete-llamada">]</span></a></sup> Esta situación se denota <span class="mwe-math-element"><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\perp y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⊥</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\perp y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36702069481de67a3e4659e380c6ac7c67d0f4ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.509ex;" alt="{\displaystyle x\perp y}"></span>. Además, un conjunto A se dice que es ortogonal a otro conjunto B, si cualquiera de los vectores de A es ortogonal a cualquiera de los vectores del conjunto B. </p> <div class="mw-heading mw-heading3"><h3 id="Complemento_ortogonal">Complemento ortogonal</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ortogonalidad_(matem%C3%A1tica)&action=edit&section=3" title="Editar sección: Complemento ortogonal"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si S es un subespacio vectorial de M, el complemento ortogonal de S en M está formado por los vectores de M que son perpendiculares a todos los vectores de S. </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Archivo:Orthogonal1.png" class="mw-file-description" title="Ejemplo 1"><img alt="Ejemplo 1" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Orthogonal1.png/120px-Orthogonal1.png" decoding="async" width="120" height="57" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Orthogonal1.png/180px-Orthogonal1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Orthogonal1.png/240px-Orthogonal1.png 2x" data-file-width="643" data-file-height="303" /></a></span></div> <div class="gallerytext">Ejemplo 1</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Archivo:Orthogonal2.png" class="mw-file-description" title="Ejemplo 2. Cálculo por el método de Gauss"><img alt="Ejemplo 2. Cálculo por el método de Gauss" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Orthogonal2.png/119px-Orthogonal2.png" decoding="async" width="119" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Orthogonal2.png/179px-Orthogonal2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Orthogonal2.png/238px-Orthogonal2.png 2x" data-file-width="489" data-file-height="493" /></a></span></div> <div class="gallerytext">Ejemplo 2. Cálculo por el método de Gauss</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Ortogonalidad_y_perpendicularidad">Ortogonalidad y perpendicularidad</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ortogonalidad_(matem%C3%A1tica)&action=edit&section=4" title="Editar sección: Ortogonalidad y perpendicularidad"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En geometría euclídea se tiene, dos <a href="/wiki/Vector_(matem%C3%A1tica)" class="mw-redirect" title="Vector (matemática)">vectores</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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </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/68baa052181f707c662844a465bfeeb135e82bab" 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> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> ortogonales forman un <a href="/wiki/%C3%81ngulo_recto" title="Ángulo recto">ángulo recto</a>, los 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 v_{1}=(3,4)}"> <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> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1}=(3,4)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e2a37210ef1f005b410f8b837f8f0d7a1d287f" 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 v_{1}=(3,4)}"></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 v_{2}=(4,-3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{2}=(4,-3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd7a204ac3fcf2895de8ec8d7d3b67bdfe8c14a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.257ex; height:2.843ex;" alt="{\displaystyle v_{2}=(4,-3)}"></span> lo son ya 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 \langle v_{1},v_{2}\rangle =v_{1}\cdot v_{2}=3\times 4+4\times (-3)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</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 fence="false" stretchy="false">⟩</mo> <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> <mn>3</mn> <mo>×</mo> <mn>4</mn> <mo>+</mo> <mn>4</mn> <mo>×</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle v_{1},v_{2}\rangle =v_{1}\cdot v_{2}=3\times 4+4\times (-3)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d9af6e00c6a80826db7af855ab4d047ce70b001" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.496ex; height:2.843ex;" alt="{\displaystyle \langle v_{1},v_{2}\rangle =v_{1}\cdot v_{2}=3\times 4+4\times (-3)=0}"></span>. En espacios no euclídeos puede definirse de modo abstracto el ángulo entre dos vectores a partir del producto interior. </p> <div class="mw-heading mw-heading3"><h3 id="Ortogonalidad_respecto_de_una_matriz_(A-ortogonalidad)"><span id="Ortogonalidad_respecto_de_una_matriz_.28A-ortogonalidad.29"></span>Ortogonalidad respecto de una matriz (A-ortogonalidad)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ortogonalidad_(matem%C3%A1tica)&action=edit&section=5" title="Editar sección: Ortogonalidad respecto de una matriz (A-ortogonalidad)"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dados dos 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 u_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69b0c788a124a32684f109737f7cfab7611d6a58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle 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 u_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2b5855eefa1e5c167320e2fb16e432c4931b166" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle u_{2}}"></span> pertenecientes a un espacio vectorial de 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"> <mi>n</mi> </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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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> y 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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" 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> de 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\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span>, si el productor escalar <span class="mwe-math-element"><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 u_{1},Au_{2}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>A</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle u_{1},Au_{2}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85d63a49666d024961470b52e379fe34b4e5d270" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.354ex; height:2.843ex;" alt="{\displaystyle \langle u_{1},Au_{2}\rangle }"></span>, notado <span class="mwe-math-element"><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 u_{1},u_{2}\rangle _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle u_{1},u_{2}\rangle _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78acd870bd7e06de914915870f1f1cd92daa0996" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.076ex; height:2.843ex;" alt="{\displaystyle \langle u_{1},u_{2}\rangle _{A}}"></span>, es igual a cero, 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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69b0c788a124a32684f109737f7cfab7611d6a58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle 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 u_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2b5855eefa1e5c167320e2fb16e432c4931b166" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle u_{2}}"></span> son ortogonales respecto a 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"> <mi>A</mi> </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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" 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> o <b>A-ortogonales</b>. Un conjunto 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"> <mi>n</mi> </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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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 \{u_{i}\}_{i=1}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{u_{i}\}_{i=1}^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89e9c814610fd14f7551ba02a2fb1f5e8f0b0371" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.354ex; height:3.009ex;" alt="{\displaystyle \{u_{i}\}_{i=1}^{n}}"></span> se dice que forma una <b>base A-ortonormal</b> 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 \langle u_{i},u_{j}\rangle _{A}=\delta _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle u_{i},u_{j}\rangle _{A}=\delta _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8afb77af36ebf2f101aa6231d1c8f03ec074f419" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.285ex; height:3.009ex;" alt="{\displaystyle \langle u_{i},u_{j}\rangle _{A}=\delta _{ij}}"></span> para todo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j=1,...,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,j=1,...,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c2a2c994bfdf328947bf99567f7898692c423be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.62ex; height:2.509ex;" alt="{\displaystyle i,j=1,...,n}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Transformación_ortogonal"><span id="Transformaci.C3.B3n_ortogonal"></span>Transformación ortogonal</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ortogonalidad_(matem%C3%A1tica)&action=edit&section=6" title="Editar sección: Transformación ortogonal"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En <a href="/wiki/Geometr%C3%ADa" title="Geometría">geometría</a> y <a href="/wiki/%C3%81lgebra_lineal" title="Álgebra lineal">álgebra lineal</a>, una transformación <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi :E\longrightarrow E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">⟶</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi :E\longrightarrow E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d8bd5a9ee0c1c17cafd69a6df3f32a35939aebc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.104ex; height:2.676ex;" alt="{\displaystyle \varphi :E\longrightarrow E}"></span> de un <a href="/wiki/Espacio_prehilbertiano" title="Espacio prehilbertiano">espacio prehilbertiano</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 (E,\langle \cdot ,\cdot \rangle )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mo fence="false" stretchy="false">⟨</mo> <mo>⋅</mo> <mo>,</mo> <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/e31a33cc421837bdb43fc2b171344bc675bd4f9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.756ex; height:2.843ex;" alt="{\displaystyle (E,\langle \cdot ,\cdot \rangle )}"></span> en sí mismo —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 \langle \cdot ,\cdot \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mo>⋅</mo> <mo>,</mo> <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/9a50080b735975d8001c9552ac2134b49ad534c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }"></span> representa el <a href="/wiki/Producto_escalar" title="Producto escalar">producto escalar</a> 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"> <mi>E</mi> </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/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" 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 ortogonal 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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> es una <a href="/wiki/Aplicaci%C3%B3n_lineal" title="Aplicación lineal">aplicación lineal</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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </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/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" 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> en sí mismo (un <a href="/wiki/Automorfismo" title="Automorfismo">automorfismo</a>) de forma que cualesquiera que sean los <span class="mwe-math-element"><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> se cumple 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 \langle \varphi (u),\varphi (v)\rangle =\langle u,v\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \varphi (u),\varphi (v)\rangle =\langle u,v\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d8ac3fba06d18cd7cb1764e5d7d8a0efc2e2fe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.358ex; height:2.843ex;" alt="{\displaystyle \langle \varphi (u),\varphi (v)\rangle =\langle u,v\rangle }"></span>. </p><p>En particular, 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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </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/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" 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> puede ser un <a href="/wiki/Espacio_eucl%C3%ADdeo" title="Espacio euclídeo">espacio euclídeo</a>. </p><p>En caso de 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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </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/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" 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> sea un <a href="/wiki/Espacio_vectorial" title="Espacio vectorial">espacio vectorial</a> sobre el <a href="/wiki/Cuerpo_(matem%C3%A1tica)" class="mw-redirect" title="Cuerpo (matemática)">cuerpo</a> de los <a href="/wiki/N%C3%BAmeros_complejos" class="mw-redirect" title="Números complejos">números complejos</a>, se dirá 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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> es <i>transformación unitaria</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Ortogonalidad_en_otros_contextos">Ortogonalidad en otros contextos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ortogonalidad_(matem%C3%A1tica)&action=edit&section=7" title="Editar sección: Ortogonalidad en otros contextos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El concepto de ortogonalidad puede extenderse a otros objetos geométricos diferente de los vectores. Por ejemplo, dos <a href="/wiki/Curva" title="Curva">curvas suaves</a> se consideran ortogonales en un punto si sus respectivos <a href="/wiki/Geometr%C3%ADa_diferencial_de_curvas#Vectores_tangente,_normal_y_binormal:_Triedro_de_Frênet-Serret" title="Geometría diferencial de curvas">vectores tangentes</a> son ortogonales. Dos <a href="/wiki/Foliaci%C3%B3n" title="Foliación">familias de curvas</a> se llaman ortogonales si en el punto de intersección de una curva de la primera familia con una curva de la segunda familia ambas resultan ser ortogonales. Un ejemplo de esto es el de las líneas isostáticas de tracción y compresión en una viga, las cuales son las <a href="/w/index.php?title=Envolventes&action=edit&redlink=1" class="new" title="Envolventes (aún no redactado)">envolventes</a> de las <a href="/wiki/Direcci%C3%B3n_principal#Tensiones_principales" title="Dirección principal">tensiones principales</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Sistemas_de_coordenadas_ortogonales">Sistemas de coordenadas ortogonales</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ortogonalidad_(matem%C3%A1tica)&action=edit&section=8" title="Editar sección: Sistemas de coordenadas ortogonales"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un <a href="/wiki/Sistema_de_coordenadas" title="Sistema de coordenadas">sistema de coordenadas</a> sobre una <a href="/wiki/Variedad_de_Riemann" title="Variedad de Riemann">variedad de Riemann</a> o un espacio <a href="/wiki/Localmente" title="Localmente">localmente</a> <a href="/wiki/Espacio_eucl%C3%ADdeo" title="Espacio euclídeo">euclídeo</a> es <b>ortogonal</b> cuando las líneas coordenadas asociadas a los valores constantes de alguna de las coordenadas tienen vectores tangentes que son ortogonales entre sí. Las <a href="/wiki/Coordenadas_cartesianas" title="Coordenadas cartesianas">coordenadas cartesianas</a>, las <a href="/wiki/Coordenadas_cil%C3%ADndricas" title="Coordenadas cilíndricas">coordenadas cilíndricas</a> y las <a href="/wiki/Coordenadas_esf%C3%A9ricas" title="Coordenadas esféricas">coordenadas esféricas</a> son ejemplos de sistemas de <a href="/wiki/Coordenadas_ortogonales" title="Coordenadas ortogonales">coordenadas ortogonales</a>. </p><p>Los sistemas de coordenadas ortogonales son interesantes porque el <a href="/wiki/Tensor_m%C3%A9trico" title="Tensor métrico">tensor métrico</a> expresado en ese sistema de coordenadas es diagonal. Si además todos los términos del tensor métrico son +1 (o también -1 si estamos en una <a href="/wiki/Variedad_pseudoriemanniana" title="Variedad pseudoriemanniana">variedad pseudoriemanniana</a>) el sistema de coordenadas se califica además de <b>ortonormal</b>. </p><p>Los sistemas de coordenadas ortogonales las líneas coordenadas forman familias de curvas ortogonales entre sí. </p> <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=Ortogonalidad_(matem%C3%A1tica)&action=edit&section=9" title="Editar sección: Véase también"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Base_(%C3%A1lgebra)" title="Base (álgebra)">Base (álgebra)</a></li> <li><a href="/wiki/Base_can%C3%B3nica" title="Base canónica">Base canónica</a></li> <li><a href="/wiki/Ortonormal" title="Ortonormal">Base Ortonormal</a></li> <li><a href="/wiki/Combinaci%C3%B3n_lineal" title="Combinación lineal">Combinación lineal</a></li> <li><a href="/wiki/Coordenadas_cartesianas" title="Coordenadas cartesianas">Coordenadas cartesianas</a></li> <li><a href="/wiki/Espacio_vectorial" title="Espacio vectorial">Espacio vectorial</a></li> <li><a href="/wiki/Independencia_lineal" class="mw-redirect" title="Independencia lineal">Independencia lineal</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> <li><a href="/wiki/Sistema_generador" title="Sistema generador">Sistema generador</a></li> <li><a href="/wiki/Vector_normal" title="Vector normal">Vector normal</a></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=Ortogonalidad_(matem%C3%A1tica)&action=edit&section=10" title="Editar sección: Referencias"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="listaref" style="list-style-type: decimal;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span class="citation web"><a rel="nofollow" class="external text" href="http://matematicas.uam.es/~fernando.chamizo/asignaturas/2021alglin/sections/4_2.pdf">«Ortogonalidad»</a>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AOrtogonalidad+%28matem%C3%A1tica%29&rft.btitle=Ortogonalidad&rft.genre=book&rft_id=http%3A%2F%2Fmatematicas.uam.es%2F~fernando.chamizo%2Fasignaturas%2F2021alglin%2Fsections%2F4_2.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <p><span id="Reference-Mathworld-Ortogonal" 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/Orthogonal.html">«Ortogonal»</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%3AOrtogonalidad+%28matem%C3%A1tica%29&rft.atitle=Ortogonal&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%2FOrthogonal.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span> </p><p><a rel="nofollow" class="external free" href="https://blog.nekomath.com/algebra-lineal-i-ortogonalidad-y-espacio-ortogonal/">https://blog.nekomath.com/algebra-lineal-i-ortogonalidad-y-espacio-ortogonal/</a> Definición y más información sobre ortogonalidad. </p> <style data-mw-deduplicate="TemplateStyles:r161257576">.mw-parser-output .mw-authority-control{margin-top:1.5em}.mw-parser-output .mw-authority-control .navbox table{margin:0}.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control .navbox+.mw-mf-linked-projects{display:none}.mw-parser-output .mw-authority-control .mw-mf-linked-projects{display:flex;padding:0.5em;border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral,#eaecf0);color:var(--color-base,#202122)}.mw-parser-output .mw-authority-control .mw-mf-linked-projects ul li{margin-bottom:0}.mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .mw-authority-control .navbox-list{border-color:#f8f9fa}.mw-parser-output .mw-authority-control .navbox th{background-color:#eeeeff}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d);background-color:var(--background-color-neutral,#27292d);color:var(--color-base,#eaecf0)}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral,#27292d)!important;color:var(--color-base,#eaecf0)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}}</style><div class="mw-authority-control"><div role="navigation" class="navbox" aria-label="Navbox" style="width: inherit;padding:3px"><table class="hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width: 12%; 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