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data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Indice</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">nascondi</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Inizio</div> </a> </li> <li id="toc-Storia" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Storia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Storia</span> </div> </a> <ul id="toc-Storia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Introduzione_elementare" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Introduzione_elementare"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Introduzione elementare</span> </div> </a> <ul id="toc-Introduzione_elementare-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nozioni_di_base" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Nozioni_di_base"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Nozioni di base</span> </div> </a> <button aria-controls="toc-Nozioni_di_base-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>Attiva/disattiva la sottosezione Nozioni di base</span> </button> <ul id="toc-Nozioni_di_base-sublist" class="vector-toc-list"> <li id="toc-Spazio_vettoriale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spazio_vettoriale"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Spazio vettoriale</span> </div> </a> <ul id="toc-Spazio_vettoriale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applicazioni_lineari" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Applicazioni_lineari"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Applicazioni lineari</span> </div> </a> <ul id="toc-Applicazioni_lineari-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Basi_e_dimensione" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Basi_e_dimensione"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Basi e dimensione</span> </div> </a> <ul id="toc-Basi_e_dimensione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prodotto_scalare" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prodotto_scalare"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Prodotto scalare</span> </div> </a> <ul id="toc-Prodotto_scalare-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applicazioni" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applicazioni"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applicazioni</span> </div> </a> <button aria-controls="toc-Applicazioni-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>Attiva/disattiva la sottosezione Applicazioni</span> </button> <ul id="toc-Applicazioni-sublist" class="vector-toc-list"> <li id="toc-Sistemi_lineari" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sistemi_lineari"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Sistemi lineari</span> </div> </a> <ul id="toc-Sistemi_lineari-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometria_analitica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometria_analitica"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Geometria analitica</span> </div> </a> <ul id="toc-Geometria_analitica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calcolo_differenziale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Calcolo_differenziale"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Calcolo differenziale</span> </div> </a> <ul id="toc-Calcolo_differenziale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Analisi_funzionale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Analisi_funzionale"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Analisi funzionale</span> </div> </a> <ul id="toc-Analisi_funzionale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Meccanica_quantistica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Meccanica_quantistica"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Meccanica quantistica</span> </div> </a> <ul id="toc-Meccanica_quantistica-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Strumenti" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Strumenti"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Strumenti</span> </div> </a> <button aria-controls="toc-Strumenti-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>Attiva/disattiva la sottosezione Strumenti</span> </button> <ul id="toc-Strumenti-sublist" class="vector-toc-list"> <li id="toc-Matrici" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matrici"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Matrici</span> </div> </a> <ul id="toc-Matrici-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eliminazione_di_Gauss" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eliminazione_di_Gauss"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Eliminazione di Gauss</span> </div> </a> <ul id="toc-Eliminazione_di_Gauss-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Determinante" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Determinante"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Determinante</span> </div> </a> <ul id="toc-Determinante-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Autovalori_e_autovettori" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Autovalori_e_autovettori"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Autovalori e autovettori</span> </div> </a> <ul id="toc-Autovalori_e_autovettori-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Forma_canonica_di_Jordan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Forma_canonica_di_Jordan"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Forma canonica di Jordan</span> </div> </a> <ul id="toc-Forma_canonica_di_Jordan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ortogonalizzazione" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ortogonalizzazione"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Ortogonalizzazione</span> </div> </a> <ul id="toc-Ortogonalizzazione-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Teoremi" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Teoremi"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Teoremi</span> </div> </a> <button aria-controls="toc-Teoremi-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>Attiva/disattiva la sottosezione Teoremi</span> </button> <ul id="toc-Teoremi-sublist" class="vector-toc-list"> <li id="toc-Teorema_della_dimensione" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Teorema_della_dimensione"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Teorema della dimensione</span> </div> </a> <ul id="toc-Teorema_della_dimensione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teorema_di_Rouché-Capelli" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Teorema_di_Rouché-Capelli"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Teorema di Rouché-Capelli</span> </div> </a> <ul id="toc-Teorema_di_Rouché-Capelli-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relazione_di_Grassmann" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relazione_di_Grassmann"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Relazione di Grassmann</span> </div> </a> <ul id="toc-Relazione_di_Grassmann-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teorema_spettrale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Teorema_spettrale"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Teorema spettrale</span> </div> </a> <ul id="toc-Teorema_spettrale-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizzazione_e_argomenti_correlati" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizzazione_e_argomenti_correlati"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Generalizzazione e argomenti correlati</span> </div> </a> <ul id="toc-Generalizzazione_e_argomenti_correlati-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Collegamenti esterni</span> </div> </a> <ul id="toc-Collegamenti_esterni-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="Indice" 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="Mostra/Nascondi l'indice" > <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">Mostra/Nascondi l'indice</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">Algebra lineare</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="Vai a una voce in un'altra lingua. Disponibile in 93 lingue" > <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-93" 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">93 lingue</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/Line%C3%AAre_algebra" title="Lineêre algebra - afrikaans" lang="af" hreflang="af" data-title="Lineêre algebra" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Lineare_Algebra" title="Lineare Algebra - tedesco svizzero" lang="gsw" hreflang="gsw" data-title="Lineare Algebra" data-language-autonym="Alemannisch" data-language-local-name="tedesco svizzero" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Alchebra_lineal" title="Alchebra lineal - aragonese" lang="an" hreflang="an" data-title="Alchebra lineal" data-language-autonym="Aragonés" data-language-local-name="aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D8%A8%D8%B1_%D8%AE%D8%B7%D9%8A" title="جبر خطي - arabo" lang="ar" hreflang="ar" data-title="جبر خطي" data-language-autonym="العربية" data-language-local-name="arabo" 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/%C3%81lxebra_llinial" title="Álxebra llinial - asturiano" lang="ast" hreflang="ast" data-title="Álxebra llinial" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/X%C9%99tti_c%C9%99br" title="Xətti cəbr - azerbaigiano" lang="az" hreflang="az" data-title="Xətti cəbr" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaigiano" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D2%BA%D1%8B%D2%99%D1%8B%D2%A1%D0%BB%D1%8B_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Һыҙыҡлы алгебра - baschiro" lang="ba" hreflang="ba" data-title="Һыҙыҡлы алгебра" data-language-autonym="Башҡортса" data-language-local-name="baschiro" 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%9B%D1%96%D0%BD%D0%B5%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Лінейная алгебра - bielorusso" lang="be" hreflang="be" data-title="Лінейная алгебра" data-language-autonym="Беларуская" data-language-local-name="bielorusso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9B%D1%96%D0%BD%D0%B5%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BB%D1%8C%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Лінейная альгебра - Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Лінейная альгебра" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</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%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Линейна алгебра - bulgaro" lang="bg" hreflang="bg" data-title="Линейна алгебра" data-language-autonym="Български" data-language-local-name="bulgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B0%E0%A7%88%E0%A6%96%E0%A6%BF%E0%A6%95_%E0%A6%AC%E0%A7%80%E0%A6%9C%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="রৈখিক বীজগণিত - bengalese" lang="bn" hreflang="bn" data-title="রৈখিক বীজগণিত" data-language-autonym="বাংলা" data-language-local-name="bengalese" 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/Linearna_algebra" title="Linearna algebra - bosniaco" lang="bs" hreflang="bs" data-title="Linearna algebra" data-language-autonym="Bosanski" data-language-local-name="bosniaco" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/%C3%80lgebra_lineal" title="Àlgebra lineal - catalano" lang="ca" hreflang="ca" data-title="Àlgebra lineal" data-language-autonym="Català" data-language-local-name="catalano" 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%AC%DB%95%D8%A8%D8%B1%DB%8C_%DA%BE%DB%8E%DA%B5%DB%8C" title="جەبری ھێڵی - curdo centrale" lang="ckb" hreflang="ckb" data-title="جەبری ھێڵی" data-language-autonym="کوردی" data-language-local-name="curdo centrale" 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/Line%C3%A1rn%C3%AD_algebra" title="Lineární algebra - ceco" lang="cs" hreflang="cs" data-title="Lineární algebra" data-language-autonym="Čeština" data-language-local-name="ceco" 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%9B%D0%B8%D0%BD%D0%B8%D0%BB%D0%BB%D0%B5_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Линилле алгебра - ciuvascio" lang="cv" hreflang="cv" data-title="Линилле алгебра" data-language-autonym="Чӑвашла" data-language-local-name="ciuvascio" 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/Algebra_llinol" title="Algebra llinol - gallese" lang="cy" hreflang="cy" data-title="Algebra llinol" data-language-autonym="Cymraeg" data-language-local-name="gallese" 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/Line%C3%A6r_algebra" title="Lineær algebra - danese" lang="da" hreflang="da" data-title="Lineær algebra" data-language-autonym="Dansk" data-language-local-name="danese" 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/Lineare_Algebra" title="Lineare Algebra - tedesco" lang="de" hreflang="de" data-title="Lineare Algebra" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-dtp mw-list-item"><a href="https://dtp.wikipedia.org/wiki/Sidtulid" title="Sidtulid - dusun centrale" lang="dtp" hreflang="dtp" data-title="Sidtulid" data-language-autonym="Kadazandusun" data-language-local-name="dusun centrale" class="interlanguage-link-target"><span>Kadazandusun</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CF%81%CE%B1%CE%BC%CE%BC%CE%B9%CE%BA%CE%AE_%CE%AC%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B1" title="Γραμμική άλγεβρα - greco" lang="el" hreflang="el" data-title="Γραμμική άλγεβρα" data-language-autonym="Ελληνικά" data-language-local-name="greco" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Linear_algebra" title="Linear algebra - inglese" lang="en" hreflang="en" data-title="Linear algebra" data-language-autonym="English" data-language-local-name="inglese" 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/Lineara_algebro" title="Lineara algebro - esperanto" lang="eo" hreflang="eo" data-title="Lineara algebro" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81lgebra_lineal" title="Álgebra lineal - spagnolo" lang="es" hreflang="es" data-title="Álgebra lineal" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Lineaaralgebra" title="Lineaaralgebra - estone" lang="et" hreflang="et" data-title="Lineaaralgebra" data-language-autonym="Eesti" data-language-local-name="estone" 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/Aljebra_lineal" title="Aljebra lineal - basco" lang="eu" hreflang="eu" data-title="Aljebra lineal" data-language-autonym="Euskara" data-language-local-name="basco" 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%AC%D8%A8%D8%B1_%D8%AE%D8%B7%DB%8C" title="جبر خطی - persiano" lang="fa" hreflang="fa" data-title="جبر خطی" data-language-autonym="فارسی" data-language-local-name="persiano" 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/Lineaarialgebra" title="Lineaarialgebra - finlandese" lang="fi" hreflang="fi" data-title="Lineaarialgebra" data-language-autonym="Suomi" data-language-local-name="finlandese" 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/Alg%C3%A8bre_lin%C3%A9aire" title="Algèbre linéaire - francese" lang="fr" hreflang="fr" data-title="Algèbre linéaire" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Lineaar_algebra" title="Lineaar algebra - frisone settentrionale" lang="frr" hreflang="frr" data-title="Lineaar algebra" data-language-autonym="Nordfriisk" data-language-local-name="frisone settentrionale" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8" title="線性代數 - gan" lang="gan" hreflang="gan" data-title="線性代數" data-language-autonym="贛語" data-language-local-name="gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Alj%C3%A8b_lin%C3%A9y%C3%A8r" title="Aljèb linéyèr - Guianan Creole" lang="gcr" hreflang="gcr" data-title="Aljèb linéyèr" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/%C3%81lxebra_lineal" title="Álxebra lineal - galiziano" lang="gl" hreflang="gl" data-title="Álxebra lineal" data-language-autonym="Galego" data-language-local-name="galiziano" 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%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="אלגברה ליניארית - ebraico" lang="he" hreflang="he" data-title="אלגברה ליניארית" data-language-autonym="עברית" data-language-local-name="ebraico" 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%B0%E0%A5%88%E0%A4%96%E0%A4%BF%E0%A4%95_%E0%A4%AC%E0%A5%80%E0%A4%9C%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="रैखिक बीजगणित - hindi" lang="hi" hreflang="hi" data-title="रैखिक बीजगणित" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Linear_algebra" title="Linear algebra - hindi figiano" lang="hif" hreflang="hif" data-title="Linear algebra" data-language-autonym="Fiji Hindi" data-language-local-name="hindi figiano" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Linearna_algebra" title="Linearna algebra - croato" lang="hr" hreflang="hr" data-title="Linearna algebra" data-language-autonym="Hrvatski" data-language-local-name="croato" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Alj%C3%A8b_liney%C3%A8" title="Aljèb lineyè - creolo haitiano" lang="ht" hreflang="ht" data-title="Aljèb lineyè" data-language-autonym="Kreyòl ayisyen" data-language-local-name="creolo haitiano" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Line%C3%A1ris_algebra" title="Lineáris algebra - ungherese" lang="hu" hreflang="hu" data-title="Lineáris algebra" data-language-autonym="Magyar" data-language-local-name="ungherese" 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/%D4%B3%D5%AE%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B0%D5%A1%D5%B6%D6%80%D5%A1%D5%B0%D5%A1%D5%B7%D5%AB%D5%BE" title="Գծային հանրահաշիվ - armeno" lang="hy" hreflang="hy" data-title="Գծային հանրահաշիվ" data-language-autonym="Հայերեն" data-language-local-name="armeno" 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/Algebra_linear" title="Algebra linear - interlingua" lang="ia" hreflang="ia" data-title="Algebra linear" 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/Aljabar_linear" title="Aljabar linear - indonesiano" lang="id" hreflang="id" data-title="Aljabar linear" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" 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/L%C3%ADnuleg_algebra" title="Línuleg algebra - islandese" lang="is" hreflang="is" data-title="Línuleg algebra" data-language-autonym="Íslenska" data-language-local-name="islandese" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B7%9A%E5%9E%8B%E4%BB%A3%E6%95%B0%E5%AD%A6" title="線型代数学 - giapponese" lang="ja" hreflang="ja" data-title="線型代数学" data-language-autonym="日本語" data-language-local-name="giapponese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Linia_haljibra" title="Linia haljibra - creolo giamaicano" lang="jam" hreflang="jam" data-title="Linia haljibra" data-language-autonym="Patois" data-language-local-name="creolo giamaicano" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%AC%E1%83%A0%E1%83%A4%E1%83%98%E1%83%95%E1%83%98_%E1%83%90%E1%83%9A%E1%83%92%E1%83%94%E1%83%91%E1%83%A0%E1%83%90" title="წრფივი ალგებრა - georgiano" lang="ka" hreflang="ka" data-title="წრფივი ალგებრა" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Aljib%E1%B9%9B_imzireg" title="Aljibṛ imzireg - cabilo" lang="kab" hreflang="kab" data-title="Aljibṛ imzireg" data-language-autonym="Taqbaylit" data-language-local-name="cabilo" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A1%D1%8B%D0%B7%D1%8B%D2%9B%D1%82%D1%8B%D2%9B_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Сызықтық алгебра - kazako" lang="kk" hreflang="kk" data-title="Сызықтық алгебра" data-language-autonym="Қазақша" data-language-local-name="kazako" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99" title="선형대수학 - coreano" lang="ko" hreflang="ko" data-title="선형대수학" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Algebra_linearis" title="Algebra linearis - latino" lang="la" hreflang="la" data-title="Algebra linearis" data-language-autonym="Latina" data-language-local-name="latino" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lez mw-list-item"><a href="https://lez.wikipedia.org/wiki/%D0%A6%D3%80%D0%B0%D1%80%D1%86%D3%80%D0%B8%D0%BD_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="ЦӀарцӀин алгебра - lesgo" lang="lez" hreflang="lez" data-title="ЦӀарцӀин алгебра" data-language-autonym="Лезги" data-language-local-name="lesgo" class="interlanguage-link-target"><span>Лезги</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Aljebra_linial" title="Aljebra linial - Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Aljebra linial" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Algebra_lineara" title="Algebra lineara - lombardo" lang="lmo" hreflang="lmo" data-title="Algebra lineara" data-language-autonym="Lombard" data-language-local-name="lombardo" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Tiesin%C4%97_algebra" title="Tiesinė algebra - lituano" lang="lt" hreflang="lt" data-title="Tiesinė algebra" 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/Line%C4%81r%C4%81_algebra" title="Lineārā algebra - lettone" lang="lv" hreflang="lv" data-title="Lineārā algebra" data-language-autonym="Latviešu" data-language-local-name="lettone" 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%9B%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Линеарна алгебра - macedone" lang="mk" hreflang="mk" data-title="Линеарна алгебра" data-language-autonym="Македонски" data-language-local-name="macedone" 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%B0%E0%B5%87%E0%B4%96%E0%B5%80%E0%B4%AF_%E0%B4%AC%E0%B5%80%E0%B4%9C%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82" title="രേഖീയ ബീജഗണിതം - malayalam" lang="ml" hreflang="ml" data-title="രേഖീയ ബീജഗണിതം" data-language-autonym="മലയാളം" data-language-local-name="malayalam" 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/Algebra_linear" title="Algebra linear - malese" lang="ms" hreflang="ms" data-title="Algebra linear" data-language-autonym="Bahasa Melayu" data-language-local-name="malese" 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/Lineaire_algebra" title="Lineaire algebra - olandese" lang="nl" hreflang="nl" data-title="Lineaire algebra" data-language-autonym="Nederlands" data-language-local-name="olandese" 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/Line%C3%A6r_algebra" title="Lineær algebra - norvegese nynorsk" lang="nn" hreflang="nn" data-title="Lineær algebra" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegese 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/Line%C3%A6r_algebra" title="Lineær algebra - norvegese bokmål" lang="nb" hreflang="nb" data-title="Lineær algebra" data-language-autonym="Norsk bokmål" data-language-local-name="norvegese bokmål" 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/Alg%C3%A8bra_lineara" title="Algèbra lineara - occitano" lang="oc" hreflang="oc" data-title="Algèbra lineara" data-language-autonym="Occitan" data-language-local-name="occitano" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Algebra_liniowa" title="Algebra liniowa - polacco" lang="pl" hreflang="pl" data-title="Algebra liniowa" data-language-autonym="Polski" data-language-local-name="polacco" 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/%C3%80lgebra_linear" title="Àlgebra linear - piemontese" lang="pms" hreflang="pms" data-title="Àlgebra linear" data-language-autonym="Piemontèis" data-language-local-name="piemontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%81lgebra_linear" title="Álgebra linear - portoghese" lang="pt" hreflang="pt" data-title="Álgebra linear" data-language-autonym="Português" data-language-local-name="portoghese" 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/Algebr%C4%83_liniar%C4%83" title="Algebră liniară - rumeno" lang="ro" hreflang="ro" data-title="Algebră liniară" data-language-autonym="Română" data-language-local-name="rumeno" 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%9B%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Линейная алгебра - russo" lang="ru" hreflang="ru" data-title="Линейная алгебра" data-language-autonym="Русский" data-language-local-name="russo" 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/Algibbra_liniari" title="Algibbra liniari - siciliano" lang="scn" hreflang="scn" data-title="Algibbra liniari" data-language-autonym="Sicilianu" data-language-local-name="siciliano" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Linear_algebra" title="Linear algebra - scozzese" lang="sco" hreflang="sco" data-title="Linear algebra" data-language-autonym="Scots" data-language-local-name="scozzese" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Linearna_algebra" title="Linearna algebra - serbo-croato" lang="sh" hreflang="sh" data-title="Linearna algebra" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croato" 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/Linear_algebra" title="Linear algebra - Simple English" lang="en-simple" hreflang="en-simple" data-title="Linear algebra" 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/Line%C3%A1rna_algebra" title="Lineárna algebra - slovacco" lang="sk" hreflang="sk" data-title="Lineárna algebra" data-language-autonym="Slovenčina" data-language-local-name="slovacco" 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/Linearna_algebra" title="Linearna algebra - sloveno" lang="sl" hreflang="sl" data-title="Linearna algebra" data-language-autonym="Slovenščina" data-language-local-name="sloveno" 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/Algjebra_lineare" title="Algjebra lineare - albanese" lang="sq" hreflang="sq" data-title="Algjebra lineare" data-language-autonym="Shqip" data-language-local-name="albanese" 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%9B%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Линеарна алгебра - serbo" lang="sr" hreflang="sr" data-title="Линеарна алгебра" data-language-autonym="Српски / srpski" data-language-local-name="serbo" 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%A4r_algebra" title="Linjär algebra - svedese" lang="sv" hreflang="sv" data-title="Linjär algebra" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Aljebra_mstari" title="Aljebra mstari - swahili" lang="sw" hreflang="sw" data-title="Aljebra mstari" data-language-autonym="Kiswahili" data-language-local-name="swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AF%87%E0%AE%B0%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%87%E0%AE%AF%E0%AE%B1%E0%AF%8D%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D" title="நேரியல் இயற்கணிதம் - tamil" lang="ta" hreflang="ta" data-title="நேரியல் இயற்கணிதம்" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8_%D1%85%D0%B0%D1%82%D1%82%D3%A3" title="Алгебраи хаттӣ - tagico" lang="tg" hreflang="tg" data-title="Алгебраи хаттӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="tagico" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9E%E0%B8%B5%E0%B8%8A%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B9%80%E0%B8%AA%E0%B9%89%E0%B8%99" title="พีชคณิตเชิงเส้น - thailandese" lang="th" hreflang="th" data-title="พีชคณิตเชิงเส้น" data-language-autonym="ไทย" data-language-local-name="thailandese" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Alhebrang_linyar" title="Alhebrang linyar - tagalog" lang="tl" hreflang="tl" data-title="Alhebrang linyar" data-language-autonym="Tagalog" data-language-local-name="tagalog" 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/Lineer_cebir" title="Lineer cebir - turco" lang="tr" hreflang="tr" data-title="Lineer cebir" 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%9B%D1%96%D0%BD%D1%96%D0%B9%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Лінійна алгебра - ucraino" lang="uk" hreflang="uk" data-title="Лінійна алгебра" data-language-autonym="Українська" data-language-local-name="ucraino" 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%AE%D8%B7%DB%8C_%D8%A7%D9%84%D8%AC%D8%A8%D8%B1%D8%A7" title="خطی الجبرا - urdu" lang="ur" hreflang="ur" data-title="خطی الجبرا" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Chiziqli_algebra" title="Chiziqli algebra - uzbeco" lang="uz" hreflang="uz" data-title="Chiziqli algebra" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeco" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BA%A1i_s%E1%BB%91_tuy%E1%BA%BFn_t%C3%ADnh" title="Đại số tuyến tính - vietnamita" lang="vi" hreflang="vi" data-title="Đại số tuyến tính" 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-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Alhebra_Linyal" title="Alhebra Linyal - waray" lang="war" hreflang="war" data-title="Alhebra Linyal" data-language-autonym="Winaray" data-language-local-name="waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="线性代数 - wu" lang="wuu" hreflang="wuu" data-title="线性代数" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%9C%D7%99%D7%A0%D7%A2%D7%90%D7%A8%D7%A2_%D7%90%D7%9C%D7%92%D7%A2%D7%91%D7%A8%D7%A2" title="לינעארע אלגעברע - yiddish" lang="yi" hreflang="yi" data-title="לינעארע אלגעברע" data-language-autonym="ייִדיש" data-language-local-name="yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/%C3%81lj%E1%BA%B9%CC%81br%C3%A0_on%C3%ADgb%E1%BB%8Dr%E1%BB%8D" title="Áljẹ́brà onígbọrọ - yoruba" lang="yo" hreflang="yo" data-title="Áljẹ́brà onígbọrọ" data-language-autonym="Yorùbá" data-language-local-name="yoruba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="线性代数 - cinese" lang="zh" hreflang="zh" data-title="线性代数" data-language-autonym="中文" data-language-local-name="cinese" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/S%C3%B2a%E2%81%BF-s%C3%A8ng_t%C4%81i-s%C3%B2%CD%98" title="Sòaⁿ-sèng tāi-sò͘ - min nan" lang="nan" hreflang="nan" data-title="Sòaⁿ-sèng tāi-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="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/%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8" title="線性代數 - cantonese" lang="yue" hreflang="yue" data-title="線性代數" data-language-autonym="粵語" data-language-local-name="cantonese" 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/Q82571#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</a></span></div> </div> </div> </div> </header> 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align="center"><span typeof="mw:File"><a href="/wiki/File:Vector_space_illust.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Vector_space_illust.svg/120px-Vector_space_illust.svg.png" decoding="async" width="120" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Vector_space_illust.svg/180px-Vector_space_illust.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Vector_space_illust.svg/240px-Vector_space_illust.svg.png 2x" data-file-width="454" data-file-height="555" /></a></span><br /><a href="/wiki/Vettore_(matematica)" title="Vettore (matematica)">Vettori</a> </td> <td align="center"><span typeof="mw:File"><a href="/wiki/File:Linear_subspaces_with_shading.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/180px-Linear_subspaces_with_shading.svg.png" decoding="async" width="180" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/270px-Linear_subspaces_with_shading.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/360px-Linear_subspaces_with_shading.svg.png 2x" data-file-width="325" data-file-height="236" /></a></span><br /><a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">Spazi vettoriali</a> </td></tr> <tr> <td align="center"><span typeof="mw:File"><a href="/wiki/File:Rotation_illustration.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Rotation_illustration.png/120px-Rotation_illustration.png" decoding="async" width="120" height="127" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Rotation_illustration.png/180px-Rotation_illustration.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Rotation_illustration.png/240px-Rotation_illustration.png 2x" data-file-width="936" data-file-height="989" /></a></span><br /><br /><a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">Trasformazioni lineari</a> </td> <td align="center"><span typeof="mw:File"><a href="/wiki/File:Intersecting_Lines.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Intersecting_Lines.svg/160px-Intersecting_Lines.svg.png" decoding="async" width="160" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Intersecting_Lines.svg/240px-Intersecting_Lines.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Intersecting_Lines.svg/320px-Intersecting_Lines.svg.png 2x" data-file-width="500" data-file-height="500" /></a></span><br /><a href="/wiki/Sistema_di_equazioni_lineari" title="Sistema di equazioni lineari">Sistemi di equazioni lineari</a> </td></tr> <tr> <td align="center"><span typeof="mw:File"><a href="/wiki/File:Matrix.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/Matrix.gif" decoding="async" width="120" height="74" class="mw-file-element" data-file-width="119" data-file-height="73" /></a></span><br /><a href="/wiki/Matrice" title="Matrice">Matrici</a> </td> <td align="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 \mathbf {A} {\color {Red}\xi }={\color {Blue}\lambda }{\color {Red}\xi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mi>ξ</mi> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#2D2F92"> <mi>λ</mi> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mi>ξ</mi> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} {\color {Red}\xi }={\color {Blue}\lambda }{\color {Red}\xi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b517ec1e28dafb02d2382c7a1d063011fd38390" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.533ex; height:2.509ex;" alt="{\displaystyle \mathbf {A} {\color {Red}\xi }={\color {Blue}\lambda }{\color {Red}\xi }}"></span><br /><a href="/wiki/Autovettore" class="mw-redirect" title="Autovettore">Autovettori e autovalori</a> </td></tr> <tr> <td align="center"><span typeof="mw:File"><a href="/wiki/File:Ellipsoide.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Ellipsoide.png/120px-Ellipsoide.png" decoding="async" width="120" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Ellipsoide.png/180px-Ellipsoide.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Ellipsoide.png/240px-Ellipsoide.png 2x" data-file-width="497" data-file-height="402" /></a></span><br /><a href="/wiki/Quadrica" title="Quadrica">Quadriche</a> </td> <td align="center"><span typeof="mw:File"><a href="/wiki/File:Epsilontensor.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Epsilontensor.svg/180px-Epsilontensor.svg.png" decoding="async" width="180" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Epsilontensor.svg/270px-Epsilontensor.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Epsilontensor.svg/360px-Epsilontensor.svg.png 2x" data-file-width="500" data-file-height="250" /></a></span><br /><a href="/wiki/Tensore" title="Tensore">Tensori</a> </td></tr></tbody></table> </td></tr></tbody></table> <p>L'<b>algebra lineare</b> è la branca della <a href="/wiki/Matematica" title="Matematica">matematica</a> che si occupa dello studio dei <a href="/wiki/Vettore_(matematica)" title="Vettore (matematica)">vettori</a>, <a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">spazi vettoriali</a> (o spazi lineari), <a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">trasformazioni lineari</a> e <a href="/wiki/Sistema_di_equazioni_lineari" title="Sistema di equazioni lineari">sistemi di equazioni lineari</a>. Gli spazi vettoriali sono un tema centrale nella <a href="/wiki/Matematica" title="Matematica">matematica</a> moderna; l'algebra lineare è usata ampiamente nell'<a href="/wiki/Algebra_astratta" title="Algebra astratta">algebra astratta</a>, nella <a href="/wiki/Geometria" title="Geometria">geometria</a> e nell'<a href="/wiki/Analisi_funzionale" title="Analisi funzionale">analisi funzionale</a>. L'algebra lineare ha inoltre una rappresentazione concreta nella <a href="/wiki/Geometria_analitica" title="Geometria analitica">geometria analitica</a>. </p><p>Con l'algebra lineare si studiano completamente tutti i fenomeni <a href="/wiki/Fisica" title="Fisica">fisici</a> "lineari", cioè quelli in cui intuitivamente non entrano in gioco <a href="/wiki/Distorsione_(fisica)" title="Distorsione (fisica)">distorsioni</a>, turbolenze e fenomeni <a href="/wiki/Teoria_del_caos" title="Teoria del caos">caotici</a> in generale. Anche fenomeni più complessi, non solo della fisica ma anche delle <a href="/wiki/Scienze_naturali" title="Scienze naturali">scienze naturali</a> e <a href="/wiki/Scienze_sociali" title="Scienze sociali">sociali</a>, possono essere studiati e ricondotti con le dovute approssimazioni a un modello lineare. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Storia">Storia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=1" title="Modifica la sezione Storia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=1" title="Edit section's source code: Storia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La storia dell'algebra lineare moderna inizia fra il <a href="/wiki/1843" title="1843">1843</a> e il <a href="/wiki/1844" title="1844">1844</a>. Nel 1843 <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a> (che ha introdotto il termine <i>vettore</i>) inventò i <a href="/wiki/Quaternione" title="Quaternione">quaternioni</a>. Nel 1844 <a href="/wiki/Hermann_Grassmann" class="mw-redirect" title="Hermann Grassmann">Hermann Grassmann</a> pubblicò il suo libro <i><a href="/w/index.php?title=Die_lineale_Ausdehnungslehre&action=edit&redlink=1" class="new" title="Die lineale Ausdehnungslehre (la pagina non esiste)">Die lineale Ausdehnungslehre</a></i>. <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a> introdusse le <a href="/wiki/Matrice" title="Matrice">matrici</a> (2×2), una delle idee fondamentali dell'algebra lineare, nel <a href="/wiki/1857" title="1857">1857</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Introduzione_elementare">Introduzione elementare</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=2" title="Modifica la sezione Introduzione elementare" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=2" title="Edit section's source code: Introduzione elementare"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File: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>Uno spazio vettoriale è una collezione di oggetti, chiamati "vettori", che possono essere sommati e riscalati.</figcaption></figure> <p>L'algebra lineare ha le sue origini nello studio dei vettori negli spazi <a href="/wiki/Coordinate_cartesiane" class="mw-redirect" title="Coordinate cartesiane">cartesiani</a> a due e a tre dimensioni. Un vettore, in questo caso, è un <a href="/wiki/Segmento" title="Segmento">segmento</a> orientato, caratterizzato da lunghezza (o magnitudine), direzione e verso. I vettori possono essere usati per rappresentare determinate entità fisiche come le <a href="/wiki/Forza" title="Forza">forze</a>, e possono essere sommati fra loro e moltiplicati per uno <a href="/wiki/Scalare_(matematica)" title="Scalare (matematica)">scalare</a>, formando quindi il primo esempio di <a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">spazio vettoriale</a> sui <a href="/wiki/Numero_reale" title="Numero reale">reali</a>. </p><p>L'algebra lineare moderna è stata estesa per comprendere spazi di dimensione arbitraria o infinita. Uno spazio vettoriale di dimensione <i>n</i> è chiamato <i>n</i>-spazio. Molti dei risultati utili nel 2-spazio e nel 3-spazio possono essere estesi agli spazi di dimensione maggiore. Anche se molte persone non sanno visualizzare facilmente i vettori negli <i>n</i>-spazi, questi vettori o <a href="/wiki/N-upla" class="mw-redirect" title="N-upla">n-uple</a> sono utili per rappresentare dati. Poiché i vettori, come <i>n</i>-uple, sono liste <i>ordinate</i> di <i>n</i> componenti, molte persone comprendono e manipolano i dati efficientemente in questa struttura. Ad esempio, in <a href="/wiki/Economia" title="Economia">economia</a>, si può creare e usare vettori 8-dimensionali (ottuple) per rappresentare il <a href="/wiki/Prodotto_Interno_Lordo" class="mw-redirect" title="Prodotto Interno Lordo">Prodotto Interno Lordo</a> di 8 stati. Si può decidere di visualizzare il PIL di 8 stati per un particolare anno, ad esempio (<a href="/wiki/Italia" title="Italia">Italia</a>, <a href="/wiki/Stati_Uniti_d%27America" title="Stati Uniti d'America">Stati Uniti</a>, <a href="/wiki/Gran_Bretagna" title="Gran Bretagna">Gran Bretagna</a>, <a href="/wiki/Francia" title="Francia">Francia</a>, <a href="/wiki/Germania" title="Germania">Germania</a>, <a href="/wiki/Spagna" title="Spagna">Spagna</a>, <a href="/wiki/Giappone" title="Giappone">Giappone</a>, <a href="/wiki/Australia" title="Australia">Australia</a>), usando un vettore (v<sub>1</sub>, v<sub>2</sub>, v<sub>3</sub>, v<sub>4</sub>, v<sub>5</sub>, v<sub>6</sub>, v<sub>7</sub>, v<sub>8</sub>) dove il PIL di ogni stato è nella sua rispettiva posizione. </p><p>Uno spazio vettoriale è definito sopra un <a href="/wiki/Campo_(matematica)" title="Campo (matematica)">campo</a>, come il campo dei <a href="/wiki/Numero_reale" title="Numero reale">numeri reali</a> o il campo dei <a href="/wiki/Numero_complesso" title="Numero complesso">numeri complessi</a>. Gli <a href="/wiki/Operatore_lineare" class="mw-redirect" title="Operatore lineare">operatori lineari</a> mappano elementi da uno spazio vettoriale su un altro (o su sé stesso), in modo che sia mantenuta la compatibilità con l'addizione e la moltiplicazione scalare definiti negli spazi vettoriali. L'insieme di tutte queste trasformazioni è anch'esso uno spazio vettoriale. Se è fissata una <a href="/wiki/Base_(algebra_lineare)" title="Base (algebra lineare)">base</a> per uno spazio vettoriale, ogni trasformazione lineare può essere rappresentata da una tabella chiamata <a href="/wiki/Matrice_(matematica)" class="mw-redirect" title="Matrice (matematica)">matrice</a>. Nell'algebra lineare si studiano quindi le proprietà delle matrici, e gli <a href="/wiki/Algoritmo" title="Algoritmo">algoritmi</a> per calcolare delle quantità importanti che le caratterizzano, quali il <a href="/wiki/Rango_(algebra_lineare)" title="Rango (algebra lineare)">rango</a>, il <a href="/wiki/Determinante_(algebra)" title="Determinante (algebra)">determinante</a> e l'insieme dei suoi <a href="/wiki/Autovalore" class="mw-redirect" title="Autovalore">autovalori</a>. </p><p>Uno spazio vettoriale (o spazio lineare), come concetto puramente astratto sul quale si provano <a href="/wiki/Teorema" title="Teorema">teoremi</a>, è parte dell'<a href="/wiki/Algebra" title="Algebra">algebra</a> astratta, e ben integrato in questo campo: alcuni oggetti algebrici correlati ad esempio sono l'<a href="/wiki/Anello_(algebra)" title="Anello (algebra)">anello</a> delle <a href="/wiki/Applicazione_lineare" class="mw-redirect" title="Applicazione lineare">mappe lineari</a> da uno spazio vettoriale in sé, o il <a href="/wiki/Gruppo_(matematica)" title="Gruppo (matematica)">gruppo</a> delle <a href="/wiki/Applicazione_lineare" class="mw-redirect" title="Applicazione lineare">mappe lineari</a> (o <a href="/wiki/Matrice_(matematica)" class="mw-redirect" title="Matrice (matematica)">matrici</a>) invertibili. L'algebra lineare gioca anche un ruolo importante in <a href="/wiki/Analisi_matematica" title="Analisi matematica">analisi</a>, specialmente nella descrizione delle <a href="/wiki/Derivata" title="Derivata">derivate</a> di ordine superiore nell'analisi vettoriale e nella risoluzione delle <a href="/wiki/Equazione_differenziale" title="Equazione differenziale">equazioni differenziali</a>. </p><p>Concludendo, si può dire semplicemente che i problemi lineari della matematica - quelli che esibiscono "linearità" nel loro comportamento - sono quelli più facili da risolvere, e che i problemi "non lineari" vengono spesso studiati approssimandoli con situazioni lineari. Ad esempio nell'<a href="/wiki/Analisi_matematica" title="Analisi matematica">analisi</a>, la <a href="/wiki/Derivata" title="Derivata">derivata</a> è un primo tentativo di <a href="/wiki/Approssimazione_lineare" title="Approssimazione lineare">approssimazione lineare</a> di una funzione. La differenza rispetto ai problemi non lineari è molto importante in pratica: il metodo generale di trovare una formulazione lineare di un problema, in termini di algebra lineare, e risolverlo, se necessario con calcoli matriciali, è uno dei metodi più generali applicabili in matematica. </p> <div class="mw-heading mw-heading2"><h2 id="Nozioni_di_base">Nozioni di base</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=3" title="Modifica la sezione Nozioni di base" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=3" title="Edit section's source code: Nozioni di base"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Spazio_vettoriale">Spazio vettoriale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=4" title="Modifica la sezione Spazio vettoriale" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=4" title="Edit section's source code: Spazio vettoriale"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r130657691">body:not(.skin-minerva) .mw-parser-output .vedi-anche{font-size:95%}</style><style data-mw-deduplicate="TemplateStyles:r139142988">.mw-parser-output .hatnote-content{align-items:center;display:flex}.mw-parser-output .hatnote-icon{flex-shrink:0}.mw-parser-output .hatnote-icon img{display:flex}.mw-parser-output .hatnote-text{font-style:italic}body:not(.skin-minerva) .mw-parser-output .hatnote{border:1px solid #CCC;display:flex;margin:.5em 0;padding:.2em .5em}body:not(.skin-minerva) .mw-parser-output .hatnote-text{padding-left:.5em}body.skin-minerva .mw-parser-output .hatnote-icon{padding-right:8px}body.skin-minerva .mw-parser-output .hatnote-icon img{height:auto;width:16px}body.skin--responsive .mw-parser-output .hatnote a.new{color:#d73333}body.skin--responsive .mw-parser-output .hatnote a.new:visited{color:#a55858}</style> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">Spazio vettoriale</a></b>.</span></div> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Scalar_multiplication_of_vectors.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Scalar_multiplication_of_vectors.svg/220px-Scalar_multiplication_of_vectors.svg.png" decoding="async" width="220" height="269" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Scalar_multiplication_of_vectors.svg/330px-Scalar_multiplication_of_vectors.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Scalar_multiplication_of_vectors.svg/440px-Scalar_multiplication_of_vectors.svg.png 2x" data-file-width="360" data-file-height="440" /></a><figcaption>Un vettore <span class="mwe-math-element"><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/ffd2487510aa438433a2579450ab2b3d557e5edc" 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> può essere riscalato, cioè moltiplicato per un numero. Qui sono mostrati i vettori <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d325c24be7d760207674a169b078892bdd5cbc76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.392ex; height:2.176ex;" alt="{\displaystyle 2a}"></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 -a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <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/6e0982b5868a66be1ed3ad7ef4bcd3d3db20f982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.038ex; height:2.176ex;" alt="{\displaystyle -a}"></span>, ottenuti moltiplicando <span class="mwe-math-element"><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/ffd2487510aa438433a2579450ab2b3d557e5edc" 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> rispettivamente per 2 e -1.</figcaption></figure> <p>La nozione più importante in algebra lineare è quella di <a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">spazio vettoriale</a>. Uno spazio vettoriale è un insieme <span class="mwe-math-element"><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> di elementi, detti <i>vettori</i>, aventi delle proprietà che li rendono simili ai vettori applicati in un punto fissato (l'<i>origine</i>) del piano o dello spazio. </p><p>Più precisamente, sono definite su <span class="mwe-math-element"><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 paio di <a href="/wiki/Operazione_binaria" title="Operazione binaria">operazioni binarie</a>:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <ul><li>due vettori <span class="mwe-math-element"><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/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle 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 w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"></span> possono essere sommati, dando così luogo ad un nuovo vettore <span class="mwe-math-element"><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+w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>+</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v+w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42b54eebfe21e642e25499aabb366f90701f9838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.632ex; height:2.176ex;" alt="{\displaystyle v+w}"></span>. Le proprietà della somma vettoriale sono <i><a href="/wiki/Associativit%C3%A0" title="Associatività">associatività</a>, <a href="/wiki/Commutativit%C3%A0" title="Commutatività">commutatività</a>, esistenza dell'<a href="/wiki/Elemento_neutro" title="Elemento neutro">elemento neutro</a>, esistenza dell'<a href="/wiki/Elemento_inverso" title="Elemento inverso">elemento inverso</a></i>;</li> <li>un vettore <span class="mwe-math-element"><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/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> può essere riscalato, cioè moltiplicato per uno <a href="/wiki/Scalare_(matematica)" title="Scalare (matematica)">scalare</a>, cioè un numero <span class="mwe-math-element"><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/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>, dando così luogo ad un nuovo vettore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle kv}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle kv}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fd84864e25712e7e2a13e23b05f52174de84a8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.339ex; height:2.176ex;" alt="{\displaystyle kv}"></span>. le proprietà della moltiplicazione per scalare sono <i>associatività, esistenza di un neutro;</i></li> <li>la somma vettoriale è <i><a href="/wiki/Distributivit%C3%A0" title="Distributività">distributiva</a> rispetto al prodotto</i>, mentre il prodotto è <i>distributivo rispetto alla somma.</i></li></ul> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Vector_addition3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Vector_addition3.svg/220px-Vector_addition3.svg.png" decoding="async" width="220" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Vector_addition3.svg/330px-Vector_addition3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Vector_addition3.svg/440px-Vector_addition3.svg.png 2x" data-file-width="190" data-file-height="78" /></a><figcaption>Due vettori <span class="mwe-math-element"><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/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle 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 w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"></span> possono essere sommati usando la <a href="/wiki/Regola_del_parallelogramma" class="mw-redirect" title="Regola del parallelogramma">regola del parallelogramma</a>.</figcaption></figure> <p>Il numero <span class="mwe-math-element"><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/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> (detto <i><a href="/wiki/Scalare_(matematica)" title="Scalare (matematica)">scalare</a></i>) appartiene ad un <a href="/wiki/Campo_(matematica)" title="Campo (matematica)">campo</a> che viene fissato fin dall'inizio: questo può essere ad esempio il campo <span class="mwe-math-element"><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> dei <a href="/wiki/Numeri_reali" class="mw-redirect" title="Numeri reali">numeri reali</a> o il campo <span class="mwe-math-element"><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> dei <a href="/wiki/Numeri_complessi" class="mw-redirect" title="Numeri complessi">numeri complessi</a>. </p><p>Il <a href="/wiki/Piano_cartesiano" class="mw-redirect" title="Piano cartesiano">piano cartesiano</a> è l'esempio fondamentale di spazio vettoriale. Ogni punto del piano è in realtà identificato univocamente come una coppia <span class="mwe-math-element"><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,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" 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 (x,y)}"></span> di numeri reali. L'origine è il punto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}"></span>. Il punto <span class="mwe-math-element"><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,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" 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 (x,y)}"></span> può essere interpretato alternativamente come punto del piano o come vettore applicato nell'origine che parte da <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}"></span> e arriva in <span class="mwe-math-element"><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,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" 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 (x,y)}"></span>. </p><p>Analogamente lo spazio cartesiano è formato da triple di punti <span class="mwe-math-element"><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,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"></span>. Più in generale, le <a href="/wiki/Ennupla" title="Ennupla">ennuple</a> di <span class="mwe-math-element"><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> numeri reali </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 (x_{1},\ldots ,x_{n})}"> <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> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},\ldots ,x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7935f7983d8a5ae59fea84efe65415235fa7c47b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle (x_{1},\ldots ,x_{n})}"></span></dd></dl> <p>formano uno spazio vettoriale che viene indicato con <span class="mwe-math-element"><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} ^{n}}"> <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"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. Ancora più in generale, si può sostituire <span class="mwe-math-element"><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> con un altro campo <span class="mwe-math-element"><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> e ottenere quindi lo spazio vettoriale <span class="mwe-math-element"><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"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </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/1d63366b3d00300e06eee81786182062b98775c5" 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>. </p> <div class="mw-heading mw-heading3"><h3 id="Applicazioni_lineari">Applicazioni lineari</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=5" title="Modifica la sezione Applicazioni lineari" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=5" title="Edit section's source code: Applicazioni lineari"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">Trasformazione lineare</a></b>.</span></div> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Rotation_illustration2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Rotation_illustration2.svg/220px-Rotation_illustration2.svg.png" decoding="async" width="220" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Rotation_illustration2.svg/330px-Rotation_illustration2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Rotation_illustration2.svg/440px-Rotation_illustration2.svg.png 2x" data-file-width="1005" data-file-height="897" /></a><figcaption>Una <a href="/wiki/Rotazione" title="Rotazione">rotazione</a> del <a href="/wiki/Piano_cartesiano" class="mw-redirect" title="Piano cartesiano">piano cartesiano</a> centrata nell'origine (0,0) è una trasformazione lineare.</figcaption></figure> <p>Un'<a href="/wiki/Applicazione_lineare" class="mw-redirect" title="Applicazione lineare">applicazione lineare</a> è una <a href="/wiki/Funzione_(matematica)" title="Funzione (matematica)">funzione</a> fra due spazi vettoriali </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:V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/574dffa1c85efaef6b6ef553ebd8ad9cf7f87fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.052ex; height:2.509ex;" alt="{\displaystyle f:V\to W}"></span></dd></dl> <p>che sia compatibile con le operazioni definite su entrambi. Devono cioè valere le proprietà seguenti: </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(v+w)=f(v)+f(w)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>+</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(v+w)=f(v)+f(w)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e21f61b8c900fc2745473cefc24cdd465784ea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.626ex; height:2.843ex;" alt="{\displaystyle f(v+w)=f(v)+f(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 f(kv)=kf(v).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(kv)=kf(v).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeacc6ad9e6e49b1d31c73e5a63200f799f539e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.599ex; height:2.843ex;" alt="{\displaystyle f(kv)=kf(v).}"></span></dd></dl> <p>per ogni coppia di vettori <span class="mwe-math-element"><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,w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>,</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v,w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6425c6e94fa47976601cb44d7564b5d04dcfbfef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.826ex; height:2.009ex;" alt="{\displaystyle v,w}"></span> in <span class="mwe-math-element"><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> e ogni scalare <span class="mwe-math-element"><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/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>. I termini "applicazione", "funzione", "trasformazione", "mappa" e "<a href="/wiki/Omomorfismo" title="Omomorfismo">omomorfismo</a>" sono in questo contesto tutti sinonimi. Il termine "lineare" sta a indicare la compatibilità con le operazioni. Un'applicazione lineare manda necessariamente l'origine (di <span class="mwe-math-element"><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>) nell'origine (di <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></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 f(0)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(0)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc3f237e89ebcbd24f17125497c63b4d3749dbf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.158ex; height:2.843ex;" alt="{\displaystyle f(0)=0.}"></span></dd></dl> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:VerticalShear_m%3D1.25.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/VerticalShear_m%3D1.25.svg/220px-VerticalShear_m%3D1.25.svg.png" decoding="async" width="220" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/VerticalShear_m%3D1.25.svg/330px-VerticalShear_m%3D1.25.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/92/VerticalShear_m%3D1.25.svg/440px-VerticalShear_m%3D1.25.svg.png 2x" data-file-width="1225" data-file-height="550" /></a><figcaption>Una trasformazione lineare può distorcere gli oggetti.</figcaption></figure> <p>Gli spazi <span class="mwe-math-element"><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> 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 W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> possono coincidere. In questo caso l'applicazione è più propriamente una <i>trasformazione</i> di <span class="mwe-math-element"><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>, ovvero una funzione che sposta i punti di <span class="mwe-math-element"><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>, chiamata anche <a href="/wiki/Endomorfismo" title="Endomorfismo">endomorfismo</a>. Una trasformazione di <span class="mwe-math-element"><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> deve necessariamente tenere fissa l'origine O. </p><p>Molte trasformazioni del <a href="/wiki/Piano_cartesiano" class="mw-redirect" title="Piano cartesiano">piano cartesiano</a> o dello spazio che tengono fissa l'origine O sono lineari: tra queste, le <a href="/wiki/Rotazione_(matematica)" title="Rotazione (matematica)">rotazioni</a> (intorno a O), le <a href="/wiki/Riflessione_(geometria)" title="Riflessione (geometria)">riflessioni</a> rispetto ad una retta o un piano (passante per O), le <a href="/wiki/Omotetia" title="Omotetia">omotetie</a> (centrate in O) e le <a href="/wiki/Proiezione_(geometria)" title="Proiezione (geometria)">proiezioni</a> (su una retta o piano passante per O). </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Squeeze_r%3D2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Squeeze_r%3D2.svg/220px-Squeeze_r%3D2.svg.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Squeeze_r%3D2.svg/330px-Squeeze_r%3D2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Squeeze_r%3D2.svg/440px-Squeeze_r%3D2.svg.png 2x" data-file-width="1110" data-file-height="556" /></a><figcaption>Una trasformazione lineare può allargare un oggetto orizzontalmente e comprimerlo verticalmente.</figcaption></figure> <p>Le applicazioni lineari compaiono in contesti molto differenti. Ad esempio, il funzionale </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 \Psi (f)=\int _{0}^{1}f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (f)=\int _{0}^{1}f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e383229276b4a3001c9bd465a36a273557aefc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.318ex; height:6.176ex;" alt="{\displaystyle \Psi (f)=\int _{0}^{1}f(x)}"></span></dd></dl> <p>che associa ad una funzione <span class="mwe-math-element"><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/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> il suo <a href="/wiki/Integrale" title="Integrale">integrale</a> è un'applicazione lineare </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 \Psi :C([0,1])\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo>:</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi :C([0,1])\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68ed84309d7b35cb6ba83d082940007174977575" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.266ex; height:2.843ex;" alt="{\displaystyle \Psi :C([0,1])\to \mathbb {R} }"></span></dd></dl> <p>dallo spazio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C([0,1])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C([0,1])}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44211c4c325ea7edb9462e7ccecda09841a41216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.228ex; height:2.843ex;" alt="{\displaystyle C([0,1])}"></span> delle <a href="/wiki/Funzione_continua" title="Funzione continua">funzioni continue</a> a valori reali in <span class="mwe-math-element"><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> <div class="mw-heading mw-heading3"><h3 id="Basi_e_dimensione">Basi e dimensione</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=6" title="Modifica la sezione Basi e dimensione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=6" title="Edit section's source code: Basi e dimensione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Base_(algebra_lineare)" title="Base (algebra lineare)">Base (algebra lineare)</a></b> e <b><a href="/wiki/Dimensione_(spazio_vettoriale)" title="Dimensione (spazio vettoriale)">Dimensione (spazio vettoriale)</a></b>.</span></div> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Basis_graph_(no_label).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Basis_graph_%28no_label%29.svg/350px-Basis_graph_%28no_label%29.svg.png" decoding="async" width="350" height="202" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Basis_graph_%28no_label%29.svg/525px-Basis_graph_%28no_label%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Basis_graph_%28no_label%29.svg/700px-Basis_graph_%28no_label%29.svg.png 2x" data-file-width="466" data-file-height="269" /></a><figcaption>I vettori <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b53cc1773694affcc1d4d6c2c778d43156a1206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (1,0)}"></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 (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span> formano la <a href="/wiki/Base_canonica" class="mw-redirect" title="Base canonica">base canonica</a> del piano cartesiano <span class="mwe-math-element"><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>. Ogni altro vettore si scrive (in modo univoco) come <a href="/wiki/Combinazione_lineare" title="Combinazione lineare">combinazione lineare</a> di questi due vettori. Ad esempio, il vettore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-2,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-2,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6d90752b9d43a223567792d6a559dd37360c7de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (-2,1)}"></span> si scrive come <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2(1,0)+1(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2(1,0)+1(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f0d92556f86e573413f52713ff67fc79c5e075e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.31ex; height:2.843ex;" alt="{\displaystyle -2(1,0)+1(0,1)}"></span>.</figcaption></figure> <p>Un punto ha dimensione zero, una retta ha dimensione uno, un piano ha dimensione due e uno spazio ha dimensione tre. L'algebra lineare permette di definire e trattare in modo rigoroso spazi di dimensione superiore alla terza. Lo spazio fondamentale di dimensione <span class="mwe-math-element"><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> è lo spazio vettoriale delle ennuple, indicato con <span class="mwe-math-element"><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} ^{n}}"> <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"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. Per <span class="mwe-math-element"><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=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02c8bd752d2cc859747ca1f3a508281bdbc3b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}"></span>, questo è l'usuale <a href="/wiki/Piano_cartesiano" class="mw-redirect" title="Piano cartesiano">piano cartesiano</a>. </p><p>Ogni spazio vettoriale <span class="mwe-math-element"><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> ha una dimensione. Questa è definita in modo algebrico, come il numero di elementi in una <a href="/wiki/Base_(algebra_lineare)" title="Base (algebra lineare)">base</a> per <span class="mwe-math-element"><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>: una base è un insieme di vettori che funge da <a href="/wiki/Sistema_di_riferimento" title="Sistema di riferimento">sistema di riferimento</a> per <span class="mwe-math-element"><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>. Rigorosamente, una base è una successione </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 {\mathcal {B}}=\{v_{1},\ldots ,v_{n}\}}"> <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-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}=\{v_{1},\ldots ,v_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c082205c3bc9f9120a510f90741b19e51e59253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.673ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}=\{v_{1},\ldots ,v_{n}\}}"></span></dd></dl> <p>di <a href="/wiki/Indipendenza_lineare" title="Indipendenza lineare">vettori indipendenti</a> che <a href="/wiki/Span_lineare" class="mw-redirect" title="Span lineare">generano</a> lo spazio <span class="mwe-math-element"><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>. Uno spazio vettoriale può avere anche dimensione infinita: gli spazi vettoriali di dimensione infinita sono spesso più complicati, e molti teoremi di algebra lineare richiedono come ipotesi che <span class="mwe-math-element"><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> abbia dimensione finita. </p><p>Le nozioni di base e dimensione si applicano 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 \mathbb {R} ^{n}}"> <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"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> (che ha dimensione <span class="mwe-math-element"><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>) e ai <a href="/wiki/Sottospazio_vettoriale" title="Sottospazio vettoriale">sottospazi</a> ivi contenuti. Essendo però definite in modo puramente algebrico, si applicano anche in contesti molto differenti: ad esempio, le <a href="/wiki/Matrice" title="Matrice">matrici</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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" 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 m\times n}"></span> formano uno spazio vettoriale di dimensione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mn}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mn}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348cd26a0b7a0034f57a951e2cf5f637dd47c1ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:1.676ex;" alt="{\displaystyle mn}"></span>. I <a href="/wiki/Polinomio" title="Polinomio">polinomi</a> in una variabile formano uno spazio vettoriale di dimensione infinita: restringendo però il grado dei polinomi ad un certo valore massimo <span class="mwe-math-element"><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> si ottiene uno spazio vettoriale di dimensione <span class="mwe-math-element"><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+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Prodotto_scalare">Prodotto scalare</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=7" title="Modifica la sezione Prodotto scalare" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=7" title="Edit section's source code: Prodotto scalare"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Prodotto_scalare" title="Prodotto scalare">Prodotto scalare</a></b>.</span></div> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Scalarproduct.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Scalarproduct.gif/220px-Scalarproduct.gif" decoding="async" width="220" height="186" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Scalarproduct.gif/330px-Scalarproduct.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Scalarproduct.gif/440px-Scalarproduct.gif 2x" data-file-width="562" data-file-height="474" /></a><figcaption>Il prodotto scalare euclideo nel piano fra due vettori <span class="mwe-math-element"><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> 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> è definito come il prodotto delle lunghezze di <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> e della proiezione di <span class="mwe-math-element"><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> su <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>. Esistono però molti altri modi utili di definire un prodotto scalare, in spazi di dimensione arbitraria.</figcaption></figure> <p>Due vettori <span class="mwe-math-element"><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/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle 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 w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"></span> di uno spazio vettoriale possono essere <i>sommati</i>: il risultato è un vettore <span class="mwe-math-element"><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+w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>+</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v+w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42b54eebfe21e642e25499aabb366f90701f9838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.632ex; height:2.176ex;" alt="{\displaystyle v+w}"></span>. Inoltre un vettore <span class="mwe-math-element"><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/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> e uno scalare <span class="mwe-math-element"><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/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> possono essere <i>moltiplicati</i>: il risultato è un vettore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle kv}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle kv}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fd84864e25712e7e2a13e23b05f52174de84a8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.339ex; height:2.176ex;" alt="{\displaystyle kv}"></span>. Nella definizione di spazio vettoriale non è però prevista un'operazione di prodotto fra due vettori. </p><p>In alcuni contesti è però utile aggiungere un'ulteriore <a href="/wiki/Operazione_binaria" title="Operazione binaria">operazione binaria</a> fra vettori, che si comporti come un prodotto. Il risultato di questo prodotto può essere a sua volta un vettore o uno scalare. Nel primo caso, questa operazione si chiama <a href="/wiki/Prodotto_vettoriale" title="Prodotto vettoriale">prodotto vettoriale</a>, e nel secondo <a href="/wiki/Prodotto_scalare" title="Prodotto scalare">prodotto scalare</a>. L'operazione di prodotto vettoriale risulta però interessante solo in dimensione tre, mentre i prodotti scalari esistono (e sono utili) in tutte le dimensioni: per questo motivo questi ultimi sono molto più studiati. </p><p>Nella definizione di spazio vettoriale non è inoltre neppure prevista una nozione di <i>lunghezza</i> (equivalentemente, <i>norma</i>) per i vettori, né di <i>angolo</i> fra due di questi. Entrambe le nozioni di lunghezza e angolo risultano però definite se è fissato un opportuno prodotto scalare.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applicazioni">Applicazioni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=8" title="Modifica la sezione Applicazioni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=8" title="Edit section's source code: Applicazioni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Sistemi_lineari">Sistemi lineari</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=9" title="Modifica la sezione Sistemi lineari" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=9" title="Edit section's source code: Sistemi lineari"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Sistema_di_equazioni_lineari" title="Sistema di equazioni lineari">Sistema di equazioni lineari</a></b>.</span></div> </div> <p>Un sistema di equazioni lineari è il dato di un certo numero <span class="mwe-math-element"><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/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> di equazioni lineari in alcune variabili <span class="mwe-math-element"><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},\ldots ,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\ldots ,x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/737e02a5fbf8bc31d443c91025339f9fd1de1065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{1},\ldots ,x_{n}}"></span>. Usando le <a href="/wiki/Matrice" title="Matrice">matrici</a> e la <a href="/wiki/Moltiplicazione_fra_matrici" class="mw-redirect" title="Moltiplicazione fra matrici">moltiplicazione riga per colonne</a>, un sistema può essere scritto in modo stringato nel modo seguente: </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 Ax=b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>=</mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax=b.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88450cdee78093e5a035c0c478a4749535b46f72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.816ex; height:2.176ex;" alt="{\displaystyle Ax=b.}"></span></dd></dl> <p>In questa espressione <span class="mwe-math-element"><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> è una matrice <span class="mwe-math-element"><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\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e173885ca0f29dd406df781144cd635276c68e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.446ex; height:2.176ex;" alt="{\displaystyle k\times 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 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/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> è il vettore delle variabili <span class="mwe-math-element"><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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" 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 x_{1}}"></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 x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ea190699149306d242b70439e663559e3ffbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}"></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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> è un altro vettore formato da costanti. </p><p>L'algebra lineare fornisce molti <a href="/wiki/Algoritmo" title="Algoritmo">algoritmi</a> per determinare le soluzioni di un sistema lineare. Il legame fra i sistemi di equazioni e l'algebra lineare sta nel fatto che la matrice <span class="mwe-math-element"><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> può essere interpretata come applicazione lineare da <span class="mwe-math-element"><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} ^{n}}"> <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"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> in <span class="mwe-math-element"><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} ^{k}}"> <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"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bcd8908c9fa46eb979ef7b67d1bb65eb3692cbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.767ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{k}}"></span>: secondo questa interpretazione, le soluzioni <span class="mwe-math-element"><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/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> sono esattamente le <a href="/wiki/Controimmagine" title="Controimmagine">controimmagini</a> di <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>. </p><p>Il <a href="/wiki/Teorema_di_Rouch%C3%A9-Capelli" title="Teorema di Rouché-Capelli">teorema di Rouché-Capelli</a> fornisce un metodo per contare le soluzioni, senza necessariamente determinarle completamente.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Nel caso in cui il sistema sia quadrato e abbia una sola soluzione, questa può essere scritta esplicitamente usando la <a href="/wiki/Regola_di_Cramer" title="Regola di Cramer">regola di Cramer</a>. Però tale soluzione teorica è praticamente utilizzabile solo per risolvere sistemi molto piccoli.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Mentre i metodi di eliminazione (es. <a href="/wiki/Metodo_di_eliminazione_di_Gauss" title="Metodo di eliminazione di Gauss">Gauss</a>) e quelli iterativi (es. <a href="/wiki/Metodo_di_Gauss-Seidel" title="Metodo di Gauss-Seidel">Gauss-Seidel</a>) consentono di calcolare effettivamente le soluzioni di un sistema lineare, anche di grandi dimensioni. </p> <div class="mw-heading mw-heading3"><h3 id="Geometria_analitica">Geometria analitica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=10" title="Modifica la sezione Geometria analitica" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=10" title="Edit section's source code: Geometria analitica"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Parallel_Lines.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Parallel_Lines.svg/220px-Parallel_Lines.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Parallel_Lines.svg/330px-Parallel_Lines.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Parallel_Lines.svg/440px-Parallel_Lines.svg.png 2x" data-file-width="1089" data-file-height="1089" /></a><figcaption>Una retta nel <a href="/wiki/Piano_cartesiano" class="mw-redirect" title="Piano cartesiano">piano cartesiano</a> è descritta da un'equazione lineare 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 ax+by+c=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax+by+c=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26ffa3f9c01bec425db7c1acc330497b6831697b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.661ex; height:2.509ex;" alt="{\displaystyle ax+by+c=0}"></span>. Due rette distinte sono parallele se il sistema formato dalle loro due equazioni non ha soluzione.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Secretsharing-3-point.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Secretsharing-3-point.png/220px-Secretsharing-3-point.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Secretsharing-3-point.png/330px-Secretsharing-3-point.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Secretsharing-3-point.png/440px-Secretsharing-3-point.png 2x" data-file-width="480" data-file-height="480" /></a><figcaption>Tre piani nello spazio possono avere varie configurazioni differenti: in questo caso si intersecano in un punto. Ciascun piano è descritto da un'equazione. Il punto di intersezione è ottenuto come soluzione di un sistema con 3 equazioni e 3 variabili.</figcaption></figure> <p>In <a href="/wiki/Geometria_analitica" title="Geometria analitica">geometria analitica</a> una retta o un piano sono descritti da sistemi di equazioni lineari: come si è appena visto, questi possono essere agevolmente studiati con gli strumenti dell'algebra lineare. Si possono quindi affrontare problemi quali le posizioni reciproche di due rette (o piani) nello spazio (che possono essere incidenti, paralleli o sghembi), e come queste variano per trasformazioni lineari. </p><p>Le <a href="/wiki/Conica" class="mw-redirect" title="Conica">coniche</a> nel piano come <a href="/wiki/Ellisse" title="Ellisse">ellisse</a>, <a href="/wiki/Parabola_(geometria)" title="Parabola (geometria)">parabola</a> e <a href="/wiki/Iperbole_(geometria)" title="Iperbole (geometria)">iperbole</a> sono determinate da <a href="/wiki/Equazione_di_secondo_grado" title="Equazione di secondo grado">equazioni di secondo grado</a>. Queste equazioni sono più complicate di quelle lineari, che sono di primo grado. Nonostante ciò, la <a href="/wiki/Classificazione_delle_coniche" class="mw-redirect" title="Classificazione delle coniche">classificazione delle coniche</a> è realizzata in modo efficace con gli strumenti dell'algebra lineare, grazie a teoremi non banali quali il <a href="/wiki/Teorema_spettrale" title="Teorema spettrale">teorema spettrale</a> e il <a href="/wiki/Teorema_di_Sylvester" title="Teorema di Sylvester">teorema di Sylvester</a>. Con gli stessi strumenti si classificano le <a href="/wiki/Quadrica" title="Quadrica">quadriche</a> nello spazio. </p> <div class="mw-heading mw-heading3"><h3 id="Calcolo_differenziale">Calcolo differenziale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=11" title="Modifica la sezione Calcolo differenziale" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=11" title="Edit section's source code: Calcolo differenziale"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'<a href="/wiki/Analisi_matematica" title="Analisi matematica">analisi matematica</a> delle funzioni in una variabile non fa uso dell'algebra lineare. L'analisi delle funzioni in più variabili invece dipende fortemente da questo settore. La nozione di <a href="/wiki/Derivata" title="Derivata">derivata</a> è infatti estesa in più variabili a quella di <a href="/wiki/Differenziale_(matematica)" title="Differenziale (matematica)">differenziale</a>: mentre la derivata è un semplice numero reale che indica la pendenza di una funzione in un punto, il differenziale è un'applicazione lineare, che indica sempre la "pendenza" di una funzione (a più variabili) in un punto. </p><p>Anche la nozione di <a href="/wiki/Derivata_seconda" class="mw-redirect" title="Derivata seconda">derivata seconda</a> si estende a più variabili: il risultato è una <a href="/wiki/Matrice" title="Matrice">matrice</a> detta <a href="/wiki/Matrice_hessiana" title="Matrice hessiana">matrice hessiana</a>. Se questa matrice è <a href="/wiki/Matrice_simmetrica" title="Matrice simmetrica">simmetrica</a>, ad esempio quando valgono le ipotesi del <a href="/wiki/Teorema_di_Schwarz" title="Teorema di Schwarz">teorema di Schwarz</a>, può essere agevolmente rappresentata come <a href="/wiki/Matrice_diagonale" title="Matrice diagonale">diagonale</a> grazie al <a href="/wiki/Teorema_spettrale" title="Teorema spettrale">teorema spettrale</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Analisi_funzionale">Analisi funzionale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=12" title="Modifica la sezione Analisi funzionale" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=12" title="Edit section's source code: Analisi funzionale"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Molti problemi dell'<a href="/wiki/Analisi_funzionale" title="Analisi funzionale">analisi funzionale</a>, quali la ricerca di una soluzione per un'<a href="/wiki/Equazione_differenziale" title="Equazione differenziale">equazione differenziale</a>, vengono affrontati analizzando un particolare <a href="/wiki/Spazio_di_funzioni" class="mw-redirect" title="Spazio di funzioni">spazio di funzioni</a>. Uno spazio di funzioni è uno <a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">spazio vettoriale</a> i cui elementi sono funzioni di un certo tipo (ad esempio continue, integrabili, derivabili... definite su un dominio fissato). Spazi di questo tipo sono generalmente di dimensione infinita, e sono dotati di alcune strutture aggiuntive, quali ad esempio un <a href="/wiki/Prodotto_scalare" title="Prodotto scalare">prodotto scalare</a> (negli <a href="/wiki/Spazio_di_Hilbert" title="Spazio di Hilbert">spazi di Hilbert</a>), una <a href="/wiki/Norma_(matematica)" title="Norma (matematica)">norma</a> (negli <a href="/wiki/Spazio_di_Banach" title="Spazio di Banach">spazi di Banach</a>) o una più generale <a href="/wiki/Spazio_topologico" title="Spazio topologico">topologia</a> (negli <a href="/wiki/Spazio_vettoriale_topologico" title="Spazio vettoriale topologico">spazi vettoriali topologici</a>). </p><p>Esempi di spazi di funzioni includono gli <a href="/wiki/Spazio_Lp" title="Spazio Lp">spazi Lp</a> e gli <a href="/wiki/Spazio_di_Sobolev" title="Spazio di Sobolev">spazi di Sobolev</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Meccanica_quantistica">Meccanica quantistica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=13" title="Modifica la sezione Meccanica quantistica" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=13" title="Edit section's source code: Meccanica quantistica"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:HAtomOrbitals.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/HAtomOrbitals.png/220px-HAtomOrbitals.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/cf/HAtomOrbitals.png 1.5x" data-file-width="316" data-file-height="316" /></a><figcaption>Le <a href="/wiki/Funzione_d%27onda" title="Funzione d'onda">funzioni d'onda</a> associate agli stati di un <a href="/wiki/Elettrone" title="Elettrone">elettrone</a> in un <a href="/wiki/Atomo_di_idrogeno" title="Atomo di idrogeno">atomo di idrogeno</a> sono gli <a href="/wiki/Autovettore" class="mw-redirect" title="Autovettore">autovettori</a> di alcuni particolari <a href="/wiki/Operatore_autoaggiunto" title="Operatore autoaggiunto">operatori autoaggiunti</a> usati in meccanica quantistica.</figcaption></figure> <p>La <a href="/wiki/Meccanica_quantistica" title="Meccanica quantistica">meccanica quantistica</a> fa ampio uso dei teoremi più avanzati dell'algebra lineare. Il <a href="/wiki/Modello_matematico" title="Modello matematico">modello matematico</a> usato in questo settore della fisica (formalizzato principalmente da <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a> e <a href="/wiki/John_Von_Neumann" class="mw-redirect" title="John Von Neumann">John Von Neumann</a>) descrive i possibili stati di un sistema quantistico come elementi di un particolare <a href="/wiki/Spazio_di_Hilbert" title="Spazio di Hilbert">spazio di Hilbert</a> e le grandezze osservabili (quali posizione, velocità, etc.) come <a href="/wiki/Operatore_autoaggiunto" title="Operatore autoaggiunto">operatori autoaggiunti</a>. I valori che possono assumere queste grandezze quando vengono effettivamente misurate sono gli <a href="/wiki/Autovalore" class="mw-redirect" title="Autovalore">autovalori</a> dell'operatore. </p><p>L'introduzione e l'uso di questi concetti matematici non banali nella fisica quantistica è stato uno dei maggiori stimoli allo sviluppo dell'algebra lineare nel <a href="/wiki/XX_secolo" title="XX secolo">XX secolo</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Strumenti">Strumenti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=14" title="Modifica la sezione Strumenti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=14" title="Edit section's source code: Strumenti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Matrici">Matrici</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=15" title="Modifica la sezione Matrici" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=15" title="Edit section's source code: Matrici"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Matrice" title="Matrice">Matrice</a></b>.</span></div> </div> <p>Una <a href="/wiki/Matrice" title="Matrice">matrice</a> è una tabella di numeri, come ad esempio: </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={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5c4b5525b9f929bc2de7efcdb5772c83ea59349" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.117ex; height:6.176ex;" alt="{\displaystyle A={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}.}"></span></dd></dl> <p>Le matrici sono utili in algebra lineare per rappresentare le applicazioni lineari. Questo viene fatto tramite la <a href="/wiki/Moltiplicazione_fra_matrici" class="mw-redirect" title="Moltiplicazione fra matrici">moltiplicazione riga per colonna</a>. Ad esempio, la matrice descritta <span class="mwe-math-element"><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> rappresenta una trasformazione del piano cartesiano <span class="mwe-math-element"><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> in sé. Questa trasformazione manda il punto <span class="mwe-math-element"><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,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" 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 (x,y)}"></span> nel punto </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{\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}-y\\x\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> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </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> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}-y\\x\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcbe029cb1261cf45c7ebbb2607c001ac9bf2d7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.356ex; height:6.176ex;" alt="{\displaystyle A{\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}-y\\x\end{pmatrix}}.}"></span></dd></dl> <p>Questa trasformazione non è nient'altro che una rotazione antioraria di 90º con centro l'origine. </p><p>La relazione fra matrici e applicazioni lineari è molto forte. Ogni applicazione lineare fra spazi vettoriali di dimensione <span class="mwe-math-element"><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"> <mi>m</mi> </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/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" 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> 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 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> è descrivibile tramite una <a href="/wiki/Matrice_associata" class="mw-redirect" title="Matrice associata">matrice associata</a> di 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 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>, purché in entrambi gli spazi vettoriali siano fissate delle <a href="/wiki/Base_(algebra_lineare)" title="Base (algebra lineare)">basi</a>. L'effetto di un cambiamento di base è codificato dalla <a href="/wiki/Matrice_di_cambiamento_di_base" title="Matrice di cambiamento di base">matrice di cambiamento di base</a> e la <a href="/wiki/Composizione_di_funzioni" title="Composizione di funzioni">composizione di funzioni</a> si traduce in una <a href="/wiki/Moltiplicazione_fra_matrici" class="mw-redirect" title="Moltiplicazione fra matrici">moltiplicazione fra matrici</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Eliminazione_di_Gauss">Eliminazione di Gauss</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=16" title="Modifica la sezione Eliminazione di Gauss" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=16" title="Edit section's source code: Eliminazione di Gauss"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Eliminazione_di_Gauss" class="mw-redirect" title="Eliminazione di Gauss">Eliminazione di Gauss</a></b>.</span></div> </div> <p>L'<a href="/wiki/Eliminazione_di_Gauss" class="mw-redirect" title="Eliminazione di Gauss">eliminazione di Gauss</a> è un algoritmo che consente di ridurre una matrice in una forma più semplice tramite opportune mosse sulle righe. Questo algoritmo è usato principalmente per determinare le soluzioni di un sistema di equazioni lineari, ma ha anche applicazioni più interne all'algebra lineare: con l'eliminazione di Gauss si può determinare il <a href="/wiki/Rango_(algebra_lineare)" title="Rango (algebra lineare)">rango</a>, il <a href="/wiki/Determinante_(algebra)" title="Determinante (algebra)">determinante</a> o l'<a href="/wiki/Matrice_inversa" class="mw-redirect" title="Matrice inversa">inversa</a> di una matrice, si può estrarre una base da un <a href="/wiki/Insieme_di_generatori" title="Insieme di generatori">insieme di generatori</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Determinante">Determinante</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=17" title="Modifica la sezione Determinante" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=17" title="Edit section's source code: Determinante"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Determinante_(algebra)" title="Determinante (algebra)">Determinante (algebra)</a></b>.</span></div> </div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Determinant_Example.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Determinant_Example.png/310px-Determinant_Example.png" decoding="async" width="310" height="175" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/c1/Determinant_Example.png 1.5x" data-file-width="446" data-file-height="252" /></a><figcaption>Una <a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">trasformazione lineare</a> del <a href="/wiki/Piano_cartesiano" class="mw-redirect" title="Piano cartesiano">piano cartesiano</a> descritta da una <a href="/wiki/Matrice_quadrata" title="Matrice quadrata">matrice quadrata</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 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}"></span>. Il determinante della matrice fornisce delle informazioni sulla trasformazione: il <a href="/wiki/Valore_assoluto" title="Valore assoluto">valore assoluto</a> descrive il cambiamento di area, mentre il segno descrive il cambiamento di <a href="/wiki/Orientazione" title="Orientazione">orientazione</a>. Nell'esempio qui riportato, la matrice ha determinante -1: quindi la trasformazione preserva le aree (un <a href="/wiki/Quadrato_(geometria)" class="mw-redirect" title="Quadrato (geometria)">quadrato</a> di area 1 si trasforma in un <a href="/wiki/Parallelogramma" title="Parallelogramma">parallelogramma</a> di area 1) ma inverte l'orientazione del piano (cambia il verso della freccia circolare).</figcaption></figure> <p>Il <a href="/wiki/Determinante_(algebra)" title="Determinante (algebra)">determinante</a> è un numero associato ad una <a href="/wiki/Matrice_quadrata" title="Matrice quadrata">matrice quadrata</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 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>, generalmente indicato come <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2d8fe180a2f848cf11e82a535b193cfe718742" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.36ex; height:2.176ex;" alt="{\displaystyle \det A}"></span>. Da un punto di vista algebrico, il determinante è importante perché vale zero precisamente quando la matrice non è <a href="/wiki/Matrice_invertibile" title="Matrice invertibile">invertibile</a>; quando non è zero, fornisce inoltre un metodo per descrivere la <a href="/wiki/Matrice_inversa" class="mw-redirect" title="Matrice inversa">matrice inversa</a> tramite la <a href="/wiki/Regola_di_Cramer" title="Regola di Cramer">regola di Cramer</a>. </p><p>Da un punto di vista geometrico, il determinante fornisce molte informazioni sulla trasformazione associata ad <span class="mwe-math-element"><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>: il suo segno (sui numeri reali) indica se la trasformazione mantiene l'<a href="/wiki/Orientazione" title="Orientazione">orientazione</a> dello spazio, ed il suo <a href="/wiki/Valore_assoluto" title="Valore assoluto">valore assoluto</a> indica come cambiano le aree degli oggetti dopo la trasformazione. </p> <div class="mw-heading mw-heading3"><h3 id="Autovalori_e_autovettori">Autovalori e autovettori</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=18" title="Modifica la sezione Autovalori e autovettori" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=18" title="Edit section's source code: Autovalori e autovettori"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Autovettore_e_autovalore" title="Autovettore e autovalore">Autovettore e autovalore</a></b>.</span></div> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Eigen.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Eigen.jpg/260px-Eigen.jpg" decoding="async" width="260" height="224" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Eigen.jpg/390px-Eigen.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Eigen.jpg/520px-Eigen.jpg 2x" data-file-width="623" data-file-height="537" /></a><figcaption>In questa trasformazione lineare della <a href="/wiki/Gioconda" title="Gioconda">Gioconda</a>, l'immagine è modificata ma l'asse centrale verticale rimane fisso. Il vettore blu ha cambiato lievemente direzione, mentre quello rosso no. Quindi il vettore rosso è un autovettore della trasformazione e quello blu no.</figcaption></figure> <p>Per caratterizzare un endomorfismo è utile studiare alcuni vettori, chiamati <a href="/wiki/Autovettore" class="mw-redirect" title="Autovettore">autovettori</a>. Geometricamente, un autovettore è un vettore che non cambia direzione. Da un punto di vista algebrico, si tratta di un vettore <span class="mwe-math-element"><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/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> tale che </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 Av=\lambda v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>v</mi> <mo>=</mo> <mi>λ</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Av=\lambda v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f7a927ca654271e6607c7f9cded57fea70c1d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.452ex; height:2.176ex;" alt="{\displaystyle Av=\lambda v}"></span></dd></dl> <p>per qualche scalare <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>, detto <a href="/wiki/Autovalore" class="mw-redirect" title="Autovalore">autovalore</a>. L'endomorfismo è particolarmente semplice da descrivere se è <a href="/wiki/Matrice_diagonalizzabile" class="mw-redirect" title="Matrice diagonalizzabile">diagonalizzabile</a>: geometricamente, questo vuol dire che esiste una base di autovettori; algebricamente, vuol dire che l'endomorfismo può essere rappresentato tramite una <a href="/wiki/Matrice_diagonale" title="Matrice diagonale">matrice diagonale</a>, come ad esempio </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}2&0\\0&-1\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> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}2&0\\0&-1\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a8af45bb6f8900a1394dd59b217593eaec608a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.275ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}2&0\\0&-1\end{pmatrix}}.}"></span></dd></dl> <p>Gli autovalori possono essere trovati agevolmente calcolando il <a href="/wiki/Polinomio_caratteristico" title="Polinomio caratteristico">polinomio caratteristico</a> della matrice associata, definito come </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 p(\lambda )=\det(A-\lambda I).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(\lambda )=\det(A-\lambda I).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6feb733f1268ac2432ccc923de027c3b1b5819cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:20.318ex; height:2.843ex;" alt="{\displaystyle p(\lambda )=\det(A-\lambda I).}"></span></dd></dl> <p>Gli autovalori di <span class="mwe-math-element"><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> sono precisamente le radici di questo polinomio. Gli autovettori si organizzano in sottospazi vettoriali chiamati <a href="/wiki/Autospazio" class="mw-redirect" title="Autospazio">autospazi</a>. Quando lo spazio vettoriale è uno spazio di funzioni, l'autovalore è una particolare funzione, chiamata <a href="/wiki/Autofunzione" title="Autofunzione">autofunzione</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Forma_canonica_di_Jordan">Forma canonica di Jordan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=19" title="Modifica la sezione Forma canonica di Jordan" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=19" title="Edit section's source code: Forma canonica di Jordan"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Forma_canonica_di_Jordan" title="Forma canonica di Jordan">Forma canonica di Jordan</a></b>.</span></div> </div> <p>Un endomorfismo diagonalizzabile è rappresentabile in modo agevole tramite una matrice diagonale. Non tutti gli endomorfismi sono però diagonalizzabili: ad esempio, una <a href="/wiki/Rotazione" title="Rotazione">rotazione</a> antioraria del piano cartesiano <span class="mwe-math-element"><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> di angolo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> è rappresentata dalla matrice </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}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \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> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab40477843fea7939707c800ffd3b668ee8ce685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.612ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{pmatrix}}}"></span></dd></dl> <p>che (per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> diverso da 0 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 \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span>) non ha autovalori né autovettori, e quindi a maggior ragione non può essere diagonalizzabile. Un altro esempio di matrice non diagonalizzabile è la seguente </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}1&1\\0&1\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> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&1\\0&1\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13ab0ffaf4c5388057dd7d41883224c4a63dd5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.467ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}1&1\\0&1\end{pmatrix}}.}"></span></dd></dl> <p>Se il campo considerato è però il campo <span class="mwe-math-element"><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> dei <a href="/wiki/Numeri_complessi" class="mw-redirect" title="Numeri complessi">numeri complessi</a>, è comunque possibile rappresentare l'endomorfismo tramite una matrice che assomiglia il più possibile ad una matrice diagonale, come nel secondo esempio mostrato. Questa rappresentazione, detta <a href="/wiki/Forma_canonica_di_Jordan" title="Forma canonica di Jordan">forma canonica di Jordan</a>, caratterizza completamente gli endomorfismi. Esiste un'analoga rappresentazione nel campo reale, leggermente più complicata. </p> <div class="mw-heading mw-heading3"><h3 id="Ortogonalizzazione">Ortogonalizzazione</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=20" title="Modifica la sezione Ortogonalizzazione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=20" title="Edit section's source code: Ortogonalizzazione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Ortogonalizzazione_di_Gram-Schmidt" title="Ortogonalizzazione di Gram-Schmidt">Ortogonalizzazione di Gram-Schmidt</a></b> e <b><a href="/wiki/Algoritmo_di_Lagrange" title="Algoritmo di Lagrange">Algoritmo di Lagrange</a></b>.</span></div> </div> <p>Similmente ad un endomorfismo, anche un prodotto scalare può essere rappresentato da una <a href="/wiki/Matrice_quadrata" title="Matrice quadrata">matrice quadrata</a>. Anche in questo contesto, la forma più semplice da trattare è quella in cui la matrice risulta essere diagonale, e questo accade precisamente quando si fissa una <a href="/wiki/Base_ortogonale" class="mw-redirect" title="Base ortogonale">base ortogonale</a>. </p><p>A differenza degli endomorfismi, in questo contesto è sempre possibile trovare una matrice diagonale. In altre parole, per qualsiasi prodotto scalare è possibile trovare basi ortogonali. Se il prodotto scalare è <a href="/wiki/Prodotto_scalare_definito_positivo" class="mw-redirect" title="Prodotto scalare definito positivo">definito positivo</a>, un algoritmo efficiente a questo scopo è l'<a href="/wiki/Ortogonalizzazione_di_Gram-Schmidt" title="Ortogonalizzazione di Gram-Schmidt">ortogonalizzazione di Gram-Schmidt</a>. Per prodotti scalari più generali si può usare l'<a href="/wiki/Algoritmo_di_Lagrange" title="Algoritmo di Lagrange">algoritmo di Lagrange</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Teoremi">Teoremi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=21" title="Modifica la sezione Teoremi" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=21" title="Edit section's source code: Teoremi"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Teorema_della_dimensione">Teorema della dimensione</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=22" title="Modifica la sezione Teorema della dimensione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=22" title="Edit section's source code: Teorema della dimensione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Teorema_della_dimensione" class="mw-redirect" title="Teorema della dimensione">Teorema della dimensione</a></b>.</span></div> </div> <p>Il <a href="/wiki/Teorema_della_dimensione" class="mw-redirect" title="Teorema della dimensione">teorema della dimensione</a> (o <i>del rango</i>) è un teorema che mette in relazione le dimensioni del <a href="/wiki/Nucleo_(matematica)" title="Nucleo (matematica)">nucleo</a> e dell'<a href="/wiki/Immagine_(matematica)" title="Immagine (matematica)">immagine</a> di un'applicazione lineare <span class="mwe-math-element"><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/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, secondo la formula: </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 \operatorname {Im} (f)+\dim \operatorname {Ker} (f)=n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>dim</mi> <mo>⁡</mo> <mi>Ker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim \operatorname {Im} (f)+\dim \operatorname {Ker} (f)=n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/349cc2ec4f16b064e5ae7af9495fdfe0c70cfe5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.208ex; height:2.843ex;" alt="{\displaystyle \dim \operatorname {Im} (f)+\dim \operatorname {Ker} (f)=n.}"></span></dd></dl> <p>Qui Im e Ker denotano immagine e nucleo, mentre <span class="mwe-math-element"><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> è la dimensione del <a href="/wiki/Dominio_(matematica)" class="mw-redirect" title="Dominio (matematica)">dominio</a> di <span class="mwe-math-element"><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/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>. Questo risultato è anche chiamato <i>teorema del rango</i>, perché tradotto nel linguaggio delle matrici assume la forma seguente: </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 \operatorname {rk} (A)+\operatorname {null} (A)=n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>rk</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>null</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {rk} (A)+\operatorname {null} (A)=n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1db297ef769aaadd0a326cd40ba0e74fbc8efa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.103ex; height:2.843ex;" alt="{\displaystyle \operatorname {rk} (A)+\operatorname {null} (A)=n.}"></span></dd></dl> <p>Qui rk e null indicano rispettivamente il <a href="/wiki/Rango_(matematica)" class="mw-redirect" title="Rango (matematica)">rango</a> e l'<a href="/wiki/Indice_di_nullit%C3%A0" class="mw-redirect" title="Indice di nullità">indice di nullità</a> di una matrice <span class="mwe-math-element"><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> 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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" 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 m\times n}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Teorema_di_Rouché-Capelli"><span id="Teorema_di_Rouch.C3.A9-Capelli"></span>Teorema di Rouché-Capelli</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=23" title="Modifica la sezione Teorema di Rouché-Capelli" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=23" title="Edit section's source code: Teorema di Rouché-Capelli"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Teorema_di_Rouch%C3%A9-Capelli" title="Teorema di Rouché-Capelli">Teorema di Rouché-Capelli</a></b>.</span></div> </div> <p>Il <a href="/wiki/Teorema_di_Rouch%C3%A9-Capelli" title="Teorema di Rouché-Capelli">teorema di Rouché-Capelli</a> dà alcune informazioni sull'insieme delle soluzioni di un generico sistema lineare: </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 Ax=b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>=</mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax=b.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88450cdee78093e5a035c0c478a4749535b46f72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.816ex; height:2.176ex;" alt="{\displaystyle Ax=b.}"></span></dd></dl> <p>Il teorema afferma due cose: </p> <ul><li>il sistema ha soluzione se e solo se le matrici <span class="mwe-math-element"><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> 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 (A|b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A|b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03ad67765373bda92dec7af9347c7bd1e664a1c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.197ex; height:2.843ex;" alt="{\displaystyle (A|b)}"></span> hanno lo stesso <a href="/wiki/Rango_(algebra_lineare)" title="Rango (algebra lineare)">rango</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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>;</li> <li>se ci sono soluzioni, queste formano un <a href="/wiki/Sottospazio_affine" title="Sottospazio affine">sottospazio affine</a> di dimensione <span class="mwe-math-element"><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-r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86c0aa8d9c52b096540edd6e6a91eb8f790a8b7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.284ex; height:2.176ex;" alt="{\displaystyle n-r}"></span>, dove <span class="mwe-math-element"><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> è il numero di incognite (cioè il numero di colonne di <span class="mwe-math-element"><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>).</li></ul> <p>In particolare, il numero di soluzioni può essere solo zero, uno o infinito (se le equazioni sono a coefficienti reali o complessi). </p><p>Il teorema è generalmente usato per determinare rapidamente se un sistema ammette una o più soluzioni: non può però essere usato per determinarle esplicitamente. Uno strumento per questo scopo è il <a href="/wiki/Metodo_di_eliminazione_di_Gauss" title="Metodo di eliminazione di Gauss">metodo di eliminazione di Gauss</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Relazione_di_Grassmann">Relazione di Grassmann</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=24" title="Modifica la sezione Relazione di Grassmann" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=24" title="Edit section's source code: Relazione di Grassmann"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Formula_di_Grassmann" title="Formula di Grassmann">Formula di Grassmann</a></b>.</span></div> </div> <p>La formula di Grassmann mette in relazione le dimensioni di vari sottospazi, definiti a partire da due sottospazi <span class="mwe-math-element"><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> 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 W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> entrambi contenuti uno spazio fissato <span class="mwe-math-element"><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>. La formula è la seguente: </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 V+\dim W=\dim(V+W)+\dim(V\cap W).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>V</mi> <mo>+</mo> <mi>dim</mi> <mo>⁡</mo> <mi>W</mi> <mo>=</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>+</mo> <mi>W</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>∩</mo> <mi>W</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim V+\dim W=\dim(V+W)+\dim(V\cap W).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52d5f5e7770547914eaf5fbc16c3225eb2a8d71b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.41ex; height:2.843ex;" alt="{\displaystyle \dim V+\dim W=\dim(V+W)+\dim(V\cap W).}"></span></dd></dl> <p>Nell'espressione compaiono i sottospazi ottenuti come somma e intersezione di <span class="mwe-math-element"><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> 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 W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Teorema_spettrale">Teorema spettrale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=25" title="Modifica la sezione Teorema spettrale" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=25" title="Edit section's source code: Teorema spettrale"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Teorema_spettrale" title="Teorema spettrale">Teorema spettrale</a></b>.</span></div> </div> <p>Il <a href="/wiki/Teorema_spettrale" title="Teorema spettrale">teorema spettrale</a> fornisce una condizione forte di <a href="/wiki/Diagonalizzabilit%C3%A0" title="Diagonalizzabilità">diagonalizzabilità</a> per alcuni endomorfismi, detti <a href="/wiki/Endomorfismo_simmetrico" class="mw-redirect" title="Endomorfismo simmetrico">simmetrici</a> o autoaggiunti (a volte il termine <i>operatore</i> è usato come sinonimo di endomorfismo). La nozione di endomorfismo simmetrico (o operatore autoaggiunto) dipende dalla presenza di un fissato <a href="/wiki/Prodotto_scalare_definito_positivo" class="mw-redirect" title="Prodotto scalare definito positivo">prodotto scalare definito positivo</a> (o <a href="/wiki/Prodotto_hermitiano" class="mw-redirect" title="Prodotto hermitiano">hermitiano</a> se si usano spazi vettoriali <a href="/wiki/Numero_complesso" title="Numero complesso">complessi</a>). Il teorema spettrale asserisce che un tale endomorfismo ha una <a href="/wiki/Base_ortonormale" title="Base ortonormale">base ortonormale</a> formata da <a href="/wiki/Autovettore" class="mw-redirect" title="Autovettore">autovettori</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizzazione_e_argomenti_correlati">Generalizzazione e argomenti correlati</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=26" title="Modifica la sezione Generalizzazione e argomenti correlati" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=26" title="Edit section's source code: Generalizzazione e argomenti correlati"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>I metodi dell'algebra lineare sono stati estesi ad altre branche della matematica, grazie al loro successo. Nella teoria dei <a href="/wiki/Modulo_(algebra)" title="Modulo (algebra)">moduli</a> si sostituisce il <a href="/wiki/Campo_(matematica)" title="Campo (matematica)">campo</a> degli scalari con un <a href="/wiki/Anello_(algebra)" title="Anello (algebra)">anello</a>. L'<a href="/w/index.php?title=Algebra_multilineare&action=edit&redlink=1" class="new" title="Algebra multilineare (la pagina non esiste)">algebra multilineare</a> si occupa dei problemi che mappano linearmente 'molte variabili' in un numero differente di variabili, portando inevitabilmente al concetto di <a href="/wiki/Tensore" title="Tensore">tensore</a>. Nella <a href="/w/index.php?title=Teoria_spettrale_degli_operatori&action=edit&redlink=1" class="new" title="Teoria spettrale degli operatori (la pagina non esiste)">teoria spettrale degli operatori</a> si riescono a gestire matrici di dimensione infinita applicando l'<a href="/wiki/Analisi_matematica" title="Analisi matematica">analisi matematica</a> in una teoria non puramente algebrica. In tutti questi casi le difficoltà tecniche sono maggiori. Inoltre, l'algebra lineare viene a essere fondamentale per ambiti riguardanti l'ottimizzazione, in particolare la <a href="/wiki/Ricerca_operativa" title="Ricerca operativa">ricerca operativa</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=27" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=27" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> Gatto, Letterio., <a rel="nofollow" class="external text" href="https://worldcat.org/oclc/956082822"><span style="font-style:italic;">Lezioni di algebra lineare e geometria per l'ingegneria : i veri appunti del corso</span></a>, CLUT, 2013, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/9788879923439" title="Speciale:RicercaISBN/9788879923439">9788879923439</a>, <a href="/wiki/Online_Computer_Library_Center" title="Online Computer Library Center">OCLC</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/oclc/956082822">956082822</a>. <small>URL consultato il 19 marzo 2019</small>.</cite></span> </li> <li id="cite_note-2"><a href="#cite_ref-2"><b>^</b></a> <span class="reference-text">Più precisamente, questo accade per uno spazio vettoriale reale dotato di un <a href="/wiki/Prodotto_scalare_definito_positivo" class="mw-redirect" title="Prodotto scalare definito positivo">prodotto scalare definito positivo</a>.</span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text">Lo spazio delle soluzioni di un sistema è uno <a href="/wiki/Spazio_affine" title="Spazio affine">spazio affine</a>, ed il teorema di Rouché-Capelli fornisce un metodo per calcolarne la dimensione. Il numero di soluzioni può essere solo 0, 1 o infinito.</span> </li> <li id="cite_note-4"><a href="#cite_ref-4"><b>^</b></a> <span class="reference-text"><a href="/wiki/Regola_di_Cramer#Problemi_nell.27applicazione" title="Regola di Cramer">Problemi computazionali che limitano l'uso della Regola di Cramer</a></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=28" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=28" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>(<span style="font-weight:bolder; font-size:80%"><abbr title="italiano">IT</abbr></span>) Ciro Ciliberto (1994): <i>Algebra lineare</i>, Boringhieri</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="italiano">IT</abbr></span>) <a href="/wiki/Serge_Lang" title="Serge Lang">Serge Lang</a> (2014): <i>Algebra lineare</i>, Boringhieri</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a href="/w/index.php?title=Steven_Roman&action=edit&redlink=1" class="new" title="Steven Roman (la pagina non esiste)">Steven Roman</a> (1992): <i>Advanced linear algebra</i>, Springer, <a href="/wiki/Speciale:RicercaISBN/0387978372" class="internal mw-magiclink-isbn">ISBN 0-387-97837-2</a></li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) de Boor, Carl, <a rel="nofollow" class="external text" href="http://digital.library.wisc.edu/1793/11635">Applied Linear Algebra</a>, (University of Wisconsin-Madison, 2002).</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Rife, Susan A, <style data-mw-deduplicate="TemplateStyles:r140554517">.mw-parser-output .chiarimento{background:#ffeaea;color:#444444}.mw-parser-output .chiarimento-apice{color:#EE0700}@media screen{html.skin-theme-clientpref-night .mw-parser-output .chiarimento{background:rgba(179,36,36,0.21);color:inherit}html.skin-theme-clientpref-night .mw-parser-output .chiarimento-apice{color:#b32424}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .chiarimento{background:rgba(179,36,36,0.21);color:inherit}html.skin-theme-clientpref-os .mw-parser-output .chiarimento-apice{color:#b32424}}</style><span class="chiarimento" title="A volte può capitare che un link presente su Wikipedia non sia più raggiungibile. Se possibile ritrova il link e inserisci il collegamento corretto, comunque non rimuovere il collegamento e inserisci il template {{Collegamento interrotto}}"><a rel="nofollow" class="external text" href="http://handle.dtic.mil/100.2/ADA316035">Matrix Algebra.</a></span><sup class="noprint chiarimento-apice" title="A volte può capitare che un link presente su Wikipedia non sia più raggiungibile. Se possibile ritrova il link e inserisci il collegamento corretto, comunque non rimuovere il collegamento e inserisci il template {{Collegamento interrotto}}">[<i><a href="/wiki/Aiuto:Collegamenti_interrotti" title="Aiuto:Collegamenti interrotti">collegamento interrotto</a></i>]</sup> (Naval Postgraduate School, Monterey, California, 1996)</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Delatorre, Anthony R. e Cooke, William K., <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r140554517"><span class="chiarimento" title="A volte può capitare che un link presente su Wikipedia non sia più raggiungibile. Se possibile ritrova il link e inserisci il collegamento corretto, comunque non rimuovere il collegamento e inserisci il template {{Collegamento interrotto}}"><a rel="nofollow" class="external text" href="http://handle.dtic.mil/100.2/ADA350689">Matrix Algebra.</a></span><sup class="noprint chiarimento-apice" title="A volte può capitare che un link presente su Wikipedia non sia più raggiungibile. Se possibile ritrova il link e inserisci il collegamento corretto, comunque non rimuovere il collegamento e inserisci il template {{Collegamento interrotto}}">[<i><a href="/wiki/Aiuto:Collegamenti_interrotti" title="Aiuto:Collegamenti interrotti">collegamento interrotto</a></i>]</sup> (Naval Postgraduate School, Monterey, California, 1998)</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Beezer, Rob, <a rel="nofollow" class="external text" href="http://linear.ups.edu/index.html"><i>A First Course in Linear Algebra</i></a>, licenza <a href="/wiki/GNU_Free_Documentation_License" title="GNU Free Documentation License">GFDL</a>.</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) J. H. M. Wedderburn <i><a rel="nofollow" class="external text" href="http://www.ams.org/online_bks/coll17/">Lectures on Matrices</a></i> (American Mathematical Society, Providence, 1934) <a href="/wiki/Speciale:RicercaISBN/0821832042" class="internal mw-magiclink-isbn">ISBN 0-8218-3204-2</a></li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Fearnley-Sander, Desmond, <a rel="nofollow" class="external text" href="http://matrix.skku.ac.kr/nla/main/CreationLA.pdf">Hermann Grassmann and the Creation of Linear Algebra</a>, American Mathematical Monthly <b>86</b> (1979), pp. 809 – 817.</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Grassmann, Hermann, <i><a rel="nofollow" class="external text" href="http://books.google.com/books?id=bKgAAAAAMAAJ">Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert</a></i>, O. Wigand, Leipzig, 1844.</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a href="/wiki/Igor%27_Rostislavovi%C4%8D_%C5%A0afarevi%C4%8D" title="Igor' Rostislavovič Šafarevič">Shafarevich, Igor R.</a>, Remizov, Alexey O. <i><a rel="nofollow" class="external text" href="https://www.springer.com/mathematics/algebra/book/978-3-642-30993-9">Linear Algebra and Geometry</a></i>, Springer, 2012, <a href="/wiki/Speciale:RicercaISBN/9783642309939" class="internal mw-magiclink-isbn">ISBN 978-3-642-30993-9</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=29" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=29" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Algoritmo_di_Lagrange" title="Algoritmo di Lagrange">Algoritmo di Lagrange</a></li> <li><a href="/wiki/Autovalore_e_autovettore" class="mw-redirect" title="Autovalore e autovettore">Autovalore e autovettore</a></li> <li><a href="/wiki/Base_(algebra_lineare)" title="Base (algebra lineare)">Base (algebra lineare)</a></li> <li><a href="/wiki/Determinante_(algebra)" title="Determinante (algebra)">Determinante</a></li> <li><a href="/wiki/Dimensione_(spazio_vettoriale)" title="Dimensione (spazio vettoriale)">Dimensione (spazio vettoriale)</a></li> <li><a href="/wiki/Eliminazione_di_Gauss" class="mw-redirect" title="Eliminazione di Gauss">Eliminazione di Gauss</a></li> <li><a href="/wiki/Forma_canonica_di_Jordan" title="Forma canonica di Jordan">Forma canonica di Jordan</a></li> <li><a href="/wiki/Formula_di_Grassmann" title="Formula di Grassmann">Formula di Grassmann</a></li> <li><a href="/wiki/Matrice" title="Matrice">Matrice</a></li> <li><a href="/wiki/Ortogonalizzazione_di_Gram-Schmidt" title="Ortogonalizzazione di Gram-Schmidt">Ortogonalizzazione di Gram-Schmidt</a></li> <li><a href="/wiki/Prodotto_scalare" title="Prodotto scalare">Prodotto scalare</a></li> <li><a href="/wiki/Sistema_di_equazioni_lineari" title="Sistema di equazioni lineari">Sistema di equazioni lineari</a></li> <li><a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">Spazio vettoriale</a></li> <li><a href="/wiki/Teorema_della_dimensione" class="mw-redirect" title="Teorema della dimensione">Teorema della dimensione</a></li> <li><a href="/wiki/Teorema_di_Rouch%C3%A9-Capelli" title="Teorema di Rouché-Capelli">Teorema di Rouché-Capelli</a></li> <li><a href="/wiki/Teorema_spettrale" title="Teorema spettrale">Teorema spettrale</a></li> <li><a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">Trasformazione lineare</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=30" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=30" title="Edit section's source code: Altri progetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em"><p id="sisterProjects" style="background-color: #efefef; color: black; font-weight: bold; margin: 0"><span>Altri progetti</span></p><ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><a href="https://it.wikibooks.org/wiki/Algebra_lineare_e_geometria_analitica" class="extiw" title="b:Algebra lineare e geometria analitica">Wikibooks</a></li> <li class="" title=""><a href="https://it.wiktionary.org/wiki/algebra_lineare" class="extiw" title="wikt:algebra lineare">Wikizionario</a></li> <li class="" title=""><a href="https://it.wikiversity.org/wiki/Materia:Algebra_lineare" class="extiw" title="v:Materia:Algebra lineare">Wikiversità</a></li> <li class="" title=""><span class="plainlinks" title="commons:Category:Linear algebra"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Linear_algebra?uselang=it">Wikimedia Commons</a></span></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://it.wikibooks.org/wiki/" title="Collabora a Wikibooks"><img alt="Collabora a Wikibooks" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/18px-Wikibooks-logo.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/27px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/36px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></a></span> <a href="https://it.wikibooks.org/wiki/" class="extiw" title="b:">Wikibooks</a> contiene testi o manuali sull'<b><a href="https://it.wikibooks.org/wiki/Algebra_lineare_e_geometria_analitica" class="extiw" title="b:Algebra lineare e geometria analitica">algebra lineare</a></b></li> <li><span typeof="mw:File"><a href="https://it.wiktionary.org/wiki/" title="Collabora a Wikizionario"><img alt="Collabora a Wikizionario" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/18px-Wiktionary_small.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/27px-Wiktionary_small.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/36px-Wiktionary_small.svg.png 2x" data-file-width="350" data-file-height="350" /></a></span> <a href="https://it.wiktionary.org/wiki/" class="extiw" title="wikt:">Wikizionario</a> contiene il lemma di dizionario «<b><a href="https://it.wiktionary.org/wiki/algebra_lineare" class="extiw" title="wikt:algebra lineare">algebra lineare</a></b>»</li> <li><span typeof="mw:File"><a href="https://it.wikiversity.org/wiki/" title="Collabora a Wikiversità"><img alt="Collabora a Wikiversità" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/18px-Wikiversity_logo_2017.svg.png" decoding="async" width="18" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/27px-Wikiversity_logo_2017.svg.png 1.5x, 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data-file-height="1376" /></a></span> <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=it">Wikimedia Commons</a></span> contiene immagini o altri file sull'<b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Linear_algebra?uselang=it">algebra lineare</a></span></b></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_lineare&veaction=edit&section=31" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Algebra_lineare&action=edit&section=31" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li class="mw-empty-elt"></li> <li><cite id="CITEREFEnciclopedia_Italiana" class="citation libro" style="font-style:normal"> Dario A. Bini e Luca Gemignani, <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/algebra-lineare_(Enciclopedia-Italiana)/"><span style="font-style:italic;">ALGEBRA LINEARE</span></a>, in <span style="font-style:italic;"><a href="/wiki/Enciclopedia_Treccani" title="Enciclopedia Treccani">Enciclopedia Italiana</a></span>, VII Appendice, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 2006.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q82571#P4223" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFSapere.it" class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.sapere.it/enciclopedia/àlgebra+lineàre.html"><span style="font-style:italic;">àlgebra lineàre</span></a>, su <span style="font-style:italic;">sapere.it</span>, <a href="/wiki/De_Agostini" title="De Agostini">De Agostini</a>.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q82571#P6706" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFEnciclopedia_della_Matematica" class="citation libro" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/algebra-lineare_(Enciclopedia-della-Matematica)/"><span style="font-style:italic;">algebra lineare</span></a>, in <span style="font-style:italic;">Enciclopedia della Matematica</span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 2013.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q82571#P9621" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFBritannica.com" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Mark Andrew Ronan, <a rel="nofollow" class="external text" href="https://www.britannica.com/topic/linear-algebra"><span style="font-style:italic;">linear algebra</span></a>, su <span style="font-style:italic;"><a href="/wiki/Enciclopedia_Britannica" title="Enciclopedia Britannica">Enciclopedia Britannica</a></span>, Encyclopædia Britannica, Inc.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q82571#P1417" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFMathWorld" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Eric W. Weisstein, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/LinearAlgebra.html"><span style="font-style:italic;">Algebra lineare</span></a>, su <span style="font-style:italic;"><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></span>, Wolfram Research.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q82571#P2812" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFSpringerEOM" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Linear_algebra"><span style="font-style:italic;">Algebra lineare</span></a>, su <span style="font-style:italic;"><a href="/wiki/Encyclopaedia_of_Mathematics" title="Encyclopaedia of Mathematics">Encyclopaedia of Mathematics</a></span>, Springer e European Mathematical Society.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q82571#P7554" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://www.math.odu.edu/~bogacki/lat/">Linear Algebra Toolkit</a>.</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://www.algebra.com/algebra/college/linear/">Linear Algebra Workbench</a>: moltiplica e inverte matrici, risolve sistemi, trova autovalori, ecc.</li></ul> <style data-mw-deduplicate="TemplateStyles:r141815314">.mw-parser-output .navbox{border:1px solid #aaa;clear:both;margin:auto;padding:2px;width:100%}.mw-parser-output .navbox th{padding-left:1em;padding-right:1em;text-align:center}.mw-parser-output .navbox>tbody>tr:first-child>th{background:#ccf;font-size:90%;width:100%;color:var(--color-base,black)}.mw-parser-output .navbox_navbar{float:left;margin:0;padding:0 10px 0 0;text-align:left;width:6em}.mw-parser-output .navbox_title{font-size:110%}.mw-parser-output .navbox_abovebelow{background:#ddf;font-size:90%;font-weight:normal}.mw-parser-output .navbox_group{background:#ddf;font-size:90%;padding:0 10px;white-space:nowrap}.mw-parser-output .navbox_list{font-size:90%;width:100%}.mw-parser-output .navbox_list a{white-space:nowrap}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_odd{background:#fdfdfd;color:var(--color-base,black)}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_even{background:#f7f7f7;color:var(--color-base,black)}.mw-parser-output .navbox a.mw-selflink{color:var(--color-base,black)}.mw-parser-output .navbox_center{text-align:center}.mw-parser-output .navbox .navbox_image{padding-left:7px;vertical-align:middle;width:0}.mw-parser-output .navbox+.navbox{margin-top:-1px}.mw-parser-output .navbox .mw-collapsible-toggle{font-weight:normal;text-align:right;width:7em}body.skin--responsive .mw-parser-output .navbox_image img{max-width:none!important}.mw-parser-output .subnavbox{margin:-3px;width:100%}.mw-parser-output .subnavbox_group{background:#e6e6ff;padding:0 10px}@media screen{html.skin-theme-clientpref-night .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-night .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-night .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-os .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-os .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}</style><table class="navbox mw-collapsible mw-collapsed noprint metadata" id="navbox-Algebra_lineare"><tbody><tr><th colspan="3" style="background:#99CCFF"><div class="navbox_navbar"><div class="noprint plainlinks" style="background-color:transparent; padding:0; font-size:xx-small; color:var(--color-base, #000000); white-space:nowrap;"><a href="/wiki/Template:Algebra_lineare" title="Template:Algebra lineare"><span title="Vai alla pagina del template">V</span></a> · <a href="/w/index.php?title=Discussioni_template:Algebra_lineare&action=edit&redlink=1" class="new" title="Discussioni template:Algebra lineare (la pagina non esiste)"><span title="Discuti del template">D</span></a> · <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Template:Algebra_lineare&action=edit"><span title="Modifica il template. Usa l'anteprima prima di salvare">M</span></a></div></div><span class="navbox_title"><a class="mw-selflink selflink">Algebra lineare</a></span></th></tr><tr><th colspan="1" class="navbox_group" style="background:#fff; text-align:right;"><a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">Spazio vettoriale</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Vettore_(matematica)" title="Vettore (matematica)">Vettore</a><b> ·</b> <a href="/wiki/Sottospazio_vettoriale" title="Sottospazio vettoriale">Sottospazio vettoriale</a> <small>(<a href="/wiki/Copertura_lineare" title="Copertura lineare">Sottospazio generato</a>)</small><b> ·</b> <a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">Applicazione lineare</a> <small>(<a href="/wiki/Nucleo_(matematica)" title="Nucleo (matematica)">Nucleo</a><b> ·</b> <a href="/wiki/Immagine_(matematica)" title="Immagine (matematica)">Immagine</a>)</small><b> ·</b> <a href="/wiki/Base_(algebra_lineare)" title="Base (algebra lineare)">Base</a><b> ·</b> <a href="/wiki/Dimensione_(spazio_vettoriale)" title="Dimensione (spazio vettoriale)">Dimensione</a><b> ·</b> <a href="/wiki/Teorema_del_rango" title="Teorema del rango">Teorema della dimensione</a><b> ·</b> <a href="/wiki/Formula_di_Grassmann" title="Formula di Grassmann">Formula di Grassmann</a><b> ·</b> <a href="/wiki/Sistema_di_equazioni_lineari" title="Sistema di equazioni lineari">Sistema lineare</a><b> ·</b> <a href="/wiki/Metodo_di_eliminazione_di_Gauss" title="Metodo di eliminazione di Gauss">Algoritmo di Gauss</a><b> ·</b> <a href="/wiki/Teorema_di_Rouch%C3%A9-Capelli" title="Teorema di Rouché-Capelli">Teorema di Rouché-Capelli</a><b> ·</b> <a href="/wiki/Regola_di_Cramer" title="Regola di Cramer">Regola di Cramer</a><b> ·</b> <a href="/wiki/Spazio_duale" title="Spazio duale">Spazio duale</a><b> ·</b> <a href="/wiki/Spazio_proiettivo" title="Spazio proiettivo">Spazio proiettivo</a><b> ·</b> <a href="/wiki/Spazio_affine" title="Spazio affine">Spazio affine</a><b> ·</b> <a href="/wiki/Teorema_della_dimensione_per_spazi_vettoriali" title="Teorema della dimensione per spazi vettoriali">Teorema della dimensione per spazi vettoriali</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#fff; text-align:right;"><a href="/wiki/Matrice" title="Matrice">Matrici</a></th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Matrice_identit%C3%A0" title="Matrice identità">Identità</a><b> ·</b> <a href="/wiki/Matrice_nulla" title="Matrice nulla">Nulla</a><b> ·</b> <a href="/wiki/Matrice_quadrata" title="Matrice quadrata">Quadrata</a><b> ·</b> <a href="/wiki/Matrice_invertibile" title="Matrice invertibile">Invertibile</a><b> ·</b> <a href="/wiki/Matrice_simmetrica" title="Matrice simmetrica">Simmetrica</a><b> ·</b> <a href="/wiki/Matrice_antisimmetrica" title="Matrice antisimmetrica">Antisimmetrica</a><b> ·</b> <a href="/wiki/Matrice_trasposta" title="Matrice trasposta">Trasposta</a><b> ·</b> <a href="/wiki/Matrice_diagonale" title="Matrice diagonale">Diagonale</a><b> ·</b> <a href="/wiki/Matrice_triangolare" title="Matrice triangolare">Triangolare</a><b> ·</b> <a href="/wiki/Matrice_di_cambiamento_di_base" title="Matrice di cambiamento di base">Di cambiamento di base</a><b> ·</b> <a href="/wiki/Matrice_ortogonale" title="Matrice ortogonale">Ortogonale</a><b> ·</b> <a href="/wiki/Matrice_normale" title="Matrice normale">Normale</a><b> ·</b> <a href="/wiki/Matrice_di_rotazione" title="Matrice di rotazione">Rotazione</a><b> ·</b> <a href="/wiki/Matrice_simplettica" title="Matrice simplettica">Simplettica</a><b> ·</b> <a href="/wiki/Moltiplicazione_di_matrici" title="Moltiplicazione di matrici">Moltiplicazione di matrici</a><b> ·</b> <a href="/wiki/Rango_(algebra_lineare)" title="Rango (algebra lineare)">Rango</a><b> ·</b> <a href="/wiki/Teorema_di_Kronecker" title="Teorema di Kronecker">Teorema di Kronecker</a><b> ·</b> <a href="/wiki/Minore_(algebra_lineare)" title="Minore (algebra lineare)">Minore</a><b> ·</b> <a href="/wiki/Matrice_dei_cofattori" title="Matrice dei cofattori">Matrice dei cofattori</a><b> ·</b> <a href="/wiki/Determinante_(algebra)" title="Determinante (algebra)">Determinante</a><b> ·</b> <a href="/wiki/Teorema_di_Binet" title="Teorema di Binet">Teorema di Binet</a><b> ·</b> <a href="/wiki/Teorema_di_Laplace" title="Teorema di Laplace">Teorema di Laplace</a><b> ·</b> <a href="/wiki/Radice_quadrata_di_una_matrice" title="Radice quadrata di una matrice">Radice quadrata di una matrice</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#fff; text-align:right;"><a href="/wiki/Diagonalizzabilit%C3%A0" title="Diagonalizzabilità">Diagonalizzabilità</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Autovettore_e_autovalore" title="Autovettore e autovalore">Autovettore e autovalore</a><b> ·</b> <a href="/wiki/Spettro_(matematica)" title="Spettro (matematica)">Spettro</a><b> ·</b> <a href="/wiki/Polinomio_caratteristico" title="Polinomio caratteristico">Polinomio caratteristico</a><b> ·</b> <a href="/wiki/Polinomio_minimo" title="Polinomio minimo">Polinomio minimo</a><b> ·</b> <a href="/wiki/Teorema_di_Hamilton-Cayley" title="Teorema di Hamilton-Cayley">Teorema di Hamilton-Cayley</a><b> ·</b> <a href="/wiki/Matrice_a_blocchi" title="Matrice a blocchi">Matrice a blocchi</a><b> ·</b> <a href="/wiki/Forma_canonica_di_Jordan" title="Forma canonica di Jordan">Forma canonica di Jordan</a><b> ·</b> <a href="/wiki/Teorema_di_diagonalizzabilit%C3%A0" title="Teorema di diagonalizzabilità">Teorema di diagonalizzabilità</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#fff; text-align:right;"><a href="/wiki/Prodotto_scalare" title="Prodotto scalare">Prodotto scalare</a></th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Forma_bilineare" title="Forma bilineare">Forma bilineare</a><b> ·</b> <a href="/wiki/Sottospazio_ortogonale" title="Sottospazio ortogonale">Sottospazio ortogonale</a><b> ·</b> <a href="/wiki/Spazio_euclideo" title="Spazio euclideo">Spazio euclideo</a><b> ·</b> <a href="/wiki/Base_ortonormale" title="Base ortonormale">Base ortonormale</a><b> ·</b> <a href="/wiki/Algoritmo_di_Lagrange" title="Algoritmo di Lagrange">Algoritmo di Lagrange</a><b> ·</b> <a href="/wiki/Segnatura_(algebra_lineare)" title="Segnatura (algebra lineare)">Segnatura</a><b> ·</b> <a href="/wiki/Teorema_di_Sylvester" title="Teorema di Sylvester">Teorema di Sylvester</a><b> ·</b> <a href="/wiki/Ortogonalizzazione_di_Gram-Schmidt" title="Ortogonalizzazione di Gram-Schmidt">Gram-Schmidt</a><b> ·</b> <a href="/wiki/Forma_sesquilineare" title="Forma sesquilineare">Forma sesquilineare</a><b> ·</b> <a href="/wiki/Forma_sesquilineare#Forma_hermitiana" title="Forma sesquilineare">Forma hermitiana</a><b> ·</b> <a href="/wiki/Teorema_spettrale" title="Teorema spettrale">Teorema spettrale</a></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141815314"><table class="navbox mw-collapsible mw-collapsed noprint metadata" id="navbox-Aree_della_matematica"><tbody><tr><th colspan="3"><div class="navbox_navbar"><div class="noprint plainlinks" style="background-color:transparent; 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href="/wiki/Aritmetica" title="Aritmetica">Aritmetica</a><b> ·</b> <a href="/wiki/Teoria_algebrica_dei_numeri" title="Teoria algebrica dei numeri">Teoria algebrica dei numeri</a><b> ·</b> <a href="/wiki/Teoria_analitica_dei_numeri" title="Teoria analitica dei numeri">Teoria analitica dei numeri</a></td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Matematica_discreta" title="Matematica discreta">Matematica discreta</a></th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Combinatoria" title="Combinatoria">Combinatoria</a><b> ·</b> <a href="/wiki/Teoria_degli_ordini" title="Teoria degli ordini">Teoria degli ordini</a><b> ·</b> <a href="/wiki/Teoria_dei_giochi" title="Teoria dei giochi">Teoria dei giochi</a></td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Geometria" title="Geometria">Geometria</a></th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Geometria_differenziale" title="Geometria differenziale">Geometria differenziale</a><b> ·</b> <a href="/wiki/Geometria_discreta" title="Geometria discreta">Geometria discreta</a><b> ·</b> <a href="/wiki/Topologia" title="Topologia">Topologia</a><b> ·</b> <a href="/wiki/Geometria_combinatoria" title="Geometria combinatoria">Geometria combinatoria</a><b> ·</b> <a href="/wiki/Geometria_algebrica" title="Geometria algebrica">Geometria algebrica</a><b> ·</b> <a href="/wiki/Teorema_di_Pitot" title="Teorema di Pitot">Teorema di Pitot</a></td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Analisi_matematica" title="Analisi matematica">Analisi matematica</a></th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Calcolo_infinitesimale" title="Calcolo infinitesimale">Calcolo infinitesimale</a><b> ·</b> <a href="/wiki/Analisi_complessa" title="Analisi complessa">Analisi complessa</a><b> ·</b> <a href="/wiki/Analisi_non_standard" title="Analisi non standard">Analisi non standard</a><b> ·</b> <a href="/wiki/Analisi_armonica" title="Analisi armonica">Analisi armonica</a><b> ·</b> <a href="/wiki/Analisi_funzionale" title="Analisi funzionale">Analisi funzionale</a><b> ·</b> <a href="/wiki/Misura_(matematica)" title="Misura (matematica)">Teoria della misura</a><b> ·</b> <a href="/wiki/Equazione_differenziale" title="Equazione differenziale">Equazioni differenziali</a><b> ·</b> <a href="/wiki/Teoria_degli_operatori" title="Teoria degli operatori">Teoria degli operatori</a></td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Teoria_della_probabilit%C3%A0" title="Teoria della probabilità">Teoria della probabilità</a> e <a href="/wiki/Statistica" title="Statistica">Statistica</a></th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Processo_stocastico" title="Processo stocastico">Processi stocastici</a></td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Fisica_matematica" title="Fisica matematica">Fisica matematica</a></th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Meccanica_classica" title="Meccanica classica">Meccanica classica</a><b> ·</b> <a href="/wiki/Sistema_dinamico" title="Sistema dinamico">Sistemi dinamici</a><b> ·</b> <a href="/wiki/Meccanica_statistica" title="Meccanica statistica">Meccanica statistica</a><b> ·</b> <a href="/wiki/Meccanica_quantistica" title="Meccanica quantistica">Meccanica quantistica</a><b> ·</b> <a href="/wiki/Meccanica_dei_fluidi" title="Meccanica dei fluidi">Meccanica dei fluidi</a></td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Analisi_numerica" title="Analisi numerica">Analisi numerica</a> e <a href="/wiki/Ricerca_operativa" title="Ricerca operativa">Ricerca operativa</a></th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Teoria_dell%27approssimazione" title="Teoria dell'approssimazione">Teoria dell'approssimazione</a><b> ·</b> <a href="/wiki/Integrazione_numerica" title="Integrazione numerica">Integrazione numerica</a><b> ·</b> <a 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title="Crittografia">Crittografia</a></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141815314"><table class="navbox mw-collapsible mw-collapsed noprint metadata" id="navbox-Geometria"><tbody><tr><th colspan="2"><div class="navbox_navbar"><div class="noprint plainlinks" style="background-color:transparent; padding:0; font-size:xx-small; color:var(--color-base, #000000); white-space:nowrap;"><a href="/wiki/Template:Geometria" title="Template:Geometria"><span title="Vai alla pagina del template">V</span></a> · <a href="/w/index.php?title=Discussioni_template:Geometria&action=edit&redlink=1" class="new" title="Discussioni template:Geometria (la pagina non esiste)"><span title="Discuti del template">D</span></a> · <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Template:Geometria&action=edit"><span title="Modifica il template. 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font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://id.loc.gov/authorities/subjects/sh85003441">sh85003441</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="tedesco">DE</abbr></span>) <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4035811-2">4035811-2</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Spagna" title="Biblioteca nazionale di Spagna">BNE</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="spagnolo">ES</abbr></span>) <a rel="nofollow" class="external text" href="http://catalogo.bne.es/uhtbin/authoritybrowse.cgi?action=display&authority_id=XX527736">XX527736</a> <a rel="nofollow" class="external text" href="http://datos.bne.es/resource/XX527736">(data)</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Francia" title="Biblioteca nazionale di Francia">BNF</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="francese">FR</abbr></span>) <a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11937509n">cb11937509n</a> <a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11937509n">(data)</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Israele" title="Biblioteca nazionale di Israele">J9U</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr>, <abbr title="ebraico">HE</abbr></span>) <a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007293931805171">987007293931805171</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_della_Dieta_nazionale_del_Giappone" title="Biblioteca della Dieta nazionale del Giappone">NDL</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr>, <abbr title="giapponese">JA</abbr></span>) <a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00570681">00570681</a></span></td></tr></tbody></table> <div class="noprint" style="width:100%; 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