Forma lineal - Viquipèdia, l'enciclopèdia lliure

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class="mw-list-item"><a href="/wiki/Especial:Discussi%C3%B3_personal" title="Discussió sobre les edicions per aquesta adreça ip. [n]" accesskey="n"><span>Discussió per aquest IP</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Lloc"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contingut" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contingut</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">amaga</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">Inici</div> </a> </li> <li id="toc-Notació" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notació"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Notació</span> </div> </a> <ul id="toc-Notació-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Objectes_duals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Objectes_duals"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Objectes duals</span> </div> </a> <ul id="toc-Objectes_duals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Càlcul" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Càlcul"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Càlcul</span> </div> </a> <ul id="toc-Càlcul-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vegeu_també" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vegeu_també"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Vegeu també</span> </div> </a> <ul id="toc-Vegeu_també-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="Contingut" 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="Commuta la taula de continguts." > <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">Commuta la taula de continguts.</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">Forma lineal</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="Vés a un article en una altra llengua. Disponible en 20 llengües" > <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-20" 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">20 llengües</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <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%AB%E0%A6%BE%E0%A6%82%E0%A6%B6%E0%A6%A8%E0%A6%BE%E0%A6%B2" title="রৈখিক ফাংশনাল - bengalí" lang="bn" hreflang="bn" data-title="রৈখিক ফাংশনাল" data-language-autonym="বাংলা" data-language-local-name="bengalí" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Line%C3%A1rn%C3%AD_funkcion%C3%A1l" title="Lineární funkcionál - txec" lang="cs" hreflang="cs" data-title="Lineární funkcionál" data-language-autonym="Čeština" data-language-local-name="txec" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Linearform" title="Linearform - alemany" lang="de" hreflang="de" data-title="Linearform" data-language-autonym="Deutsch" data-language-local-name="alemany" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Linear_form" title="Linear form - anglès" lang="en" hreflang="en" data-title="Linear form" data-language-autonym="English" data-language-local-name="anglès" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es badge-Q70893996 mw-list-item" title=""><a href="https://es.wikipedia.org/wiki/Funcional_lineal" title="Funcional lineal - espanyol" lang="es" hreflang="es" data-title="Funcional lineal" data-language-autonym="Español" data-language-local-name="espanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AD%D8%A7%D9%84%D8%AA_%D8%AE%D8%B7%DB%8C" title="حالت خطی - persa" lang="fa" hreflang="fa" data-title="حالت خطی" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Forme_lin%C3%A9aire" title="Forme linéaire - francès" lang="fr" hreflang="fr" data-title="Forme linéaire" data-language-autonym="Français" data-language-local-name="francès" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Line%C3%A1ris_forma" title="Lineáris forma - hongarès" lang="hu" hreflang="hu" data-title="Lineáris forma" data-language-autonym="Magyar" data-language-local-name="hongarès" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzionale_lineare" title="Funzionale lineare - italià" lang="it" hreflang="it" data-title="Funzionale lineare" data-language-autonym="Italiano" data-language-local-name="italià" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B7%9A%E5%9E%8B%E6%B1%8E%E5%87%BD%E6%95%B0" title="線型汎函数 - japonès" lang="ja" hreflang="ja" data-title="線型汎函数" data-language-autonym="日本語" data-language-local-name="japonès" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lineaire_functionaal" title="Lineaire functionaal - neerlandès" lang="nl" hreflang="nl" data-title="Lineaire functionaal" data-language-autonym="Nederlands" data-language-local-name="neerlandès" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Line%C3%A6r_funksjonal" title="Lineær funksjonal - noruec bokmål" lang="nb" hreflang="nb" data-title="Lineær funksjonal" data-language-autonym="Norsk bokmål" data-language-local-name="noruec bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B2%E0%A9%80%E0%A8%A8%E0%A9%80%E0%A8%85%E0%A8%B0_%E0%A8%85%E0%A8%95%E0%A8%BE%E0%A8%B0" title="ਲੀਨੀਅਰ ਅਕਾਰ - panjabi" lang="pa" hreflang="pa" data-title="ਲੀਨੀਅਰ ਅਕਾਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Forma_liniowa" title="Forma liniowa - polonès" lang="pl" hreflang="pl" data-title="Forma liniowa" data-language-autonym="Polski" data-language-local-name="polonès" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Forma_linear" title="Forma linear - portuguès" lang="pt" hreflang="pt" data-title="Forma linear" data-language-autonym="Português" data-language-local-name="portuguès" class="interlanguage-link-target"><span>Português</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_%D1%84%D0%BE%D1%80%D0%BC%D0%B0" title="Линейная форма - rus" lang="ru" hreflang="ru" data-title="Линейная форма" data-language-autonym="Русский" data-language-local-name="rus" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Line%C3%A1rny_funkcion%C3%A1l" title="Lineárny funkcionál - eslovac" lang="sk" hreflang="sk" data-title="Lineárny funkcionál" data-language-autonym="Slovenčina" data-language-local-name="eslovac" class="interlanguage-link-target"><span>Slovenčina</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_%D1%84%D0%BE%D1%80%D0%BC%D0%B0" title="Лінійна форма - ucraïnès" lang="uk" hreflang="uk" data-title="Лінійна форма" data-language-autonym="Українська" data-language-local-name="ucraïnès" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Phi%E1%BA%BFm_h%C3%A0m_tuy%E1%BA%BFn_t%C3%ADnh" title="Phiếm hàm tuyến tính - vietnamita" lang="vi" hreflang="vi" data-title="Phiếm hàm 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-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%B7%9A%E6%80%A7%E6%B3%9B%E5%87%BD" title="線性泛函 - xinès" lang="zh" hreflang="zh" data-title="線性泛函" data-language-autonym="中文" data-language-local-name="xinès" class="interlanguage-link-target"><span>中文</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/Q261527#sitelinks-wikipedia" title="Modifica enllaços interlingües" class="wbc-editpage">Modifica els enllaços</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espais de noms"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Forma_lineal" title="Vegeu el contingut de la pàgina [c]" accesskey="c"><span>Pàgina</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Discussi%C3%B3:Forma_lineal&action=edit&redlink=1" rel="discussion" class="new" title="Discussió sobre el contingut d'aquesta pàgina (encara no existeix) [t]" accesskey="t"><span>Discussió</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Canvia la variant de llengua" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">català</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Vistes"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Forma_lineal"><span>Mostra</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Forma_lineal&action=edit" title="Modifica el codi font d'aquesta pàgina [e]" accesskey="e"><span>Modifica</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Forma_lineal&action=history" title="Versions antigues d'aquesta pàgina [h]" accesskey="h"><span>Mostra l'historial</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Eines de la pàgina"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Eines" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Eines</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Eines</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">amaga</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Més opcions" > <div class="vector-menu-heading"> Accions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Forma_lineal"><span>Mostra</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Forma_lineal&action=edit" title="Modifica el codi font d'aquesta pàgina [e]" accesskey="e"><span>Modifica</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Forma_lineal&action=history"><span>Mostra l'historial</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:Enlla%C3%A7os/Forma_lineal" title="Una llista de totes les pàgines wiki que enllacen amb aquesta [j]" accesskey="j"><span>Què hi enllaça</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:Seguiment/Forma_lineal" rel="nofollow" title="Canvis recents a pàgines enllaçades des d'aquesta pàgina [k]" accesskey="k"><span>Canvis relacionats</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Especial:P%C3%A0gines_especials" title="Llista totes les pàgines especials [q]" accesskey="q"><span>Pàgines especials</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Forma_lineal&oldid=30823900" title="Enllaç permanent a aquesta revisió de la pàgina"><span>Enllaç permanent</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Forma_lineal&action=info" title="Més informació sobre aquesta pàgina"><span>Informació de la pàgina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citau&page=Forma_lineal&id=30823900&wpFormIdentifier=titleform" title="Informació sobre com citar aquesta pàgina"><span>Citau aquest article</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:UrlShortener&url=https%3A%2F%2Fca.wikipedia.org%2Fwiki%2FForma_lineal"><span>Obtén una URL abreujada</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrCode&url=https%3A%2F%2Fca.wikipedia.org%2Fwiki%2FForma_lineal"><span>Descarrega el codi QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Imprimeix/exporta </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Especial:Llibre&bookcmd=book_creator&referer=Forma+lineal"><span>Crea un llibre</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=Forma_lineal&action=show-download-screen"><span>Baixa com a PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Forma_lineal&printable=yes" title="Versió per a impressió d'aquesta pàgina [p]" accesskey="p"><span>Versió per a impressora</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> En altres projectes </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q261527" title="Enllaç a l'element del repositori de dades connectat [g]" accesskey="g"><span>Element a Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Eines de la pàgina"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aparença"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header 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id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ca" dir="ltr"><p>Sigui <i>V</i> un objecte matemàtic qualsevol amb estructura lineal sobre un altre objecte <i>K</i> amb estructura aritmètica. Típicament <i>V</i> és un <i>K</i>-<a href="/wiki/M%C3%B2dul" title="Mòdul">mòdul</a> sobre un <a href="/wiki/Anell_(matem%C3%A0tiques)" title="Anell (matemàtiques)">anell</a> <i>K</i>, o un <a href="/wiki/Espai_vectorial" title="Espai vectorial">espai vectorial</a> sobre un <a href="/wiki/Cos_(matem%C3%A0tiques)" title="Cos (matemàtiques)">cos</a> <i>K</i>. Una <b>forma lineal</b> és una <a href="/wiki/Funci%C3%B3_matem%C3%A0tica" class="mw-redirect" title="Funció matemàtica">aplicació</a> </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 \omega :V\longrightarrow K\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">⟶</mo> <mi>K</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega :V\longrightarrow K\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf2bd7b8facc581dbbce63b62cc72a4afe1186cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.719ex; height:2.176ex;" alt="{\displaystyle \omega :V\longrightarrow K\,}"></span></dd></dl> <p>de l'objecte <i>V</i> a l'objecte <i>K</i> que compleix el requeriment de linealitat: </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 \omega (\lambda x+\mu y)=\lambda \omega (x)+\mu \omega (y),\quad x,y\in V,\quad \lambda ,\mu \in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>x</mi> <mo>+</mo> <mi>μ</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>μ</mi> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>V</mi> <mo>,</mo> <mspace width="1em" /> <mi>λ</mi> <mo>,</mo> <mi>μ</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (\lambda x+\mu y)=\lambda \omega (x)+\mu \omega (y),\quad x,y\in V,\quad \lambda ,\mu \in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bec31edaf7dd481cd06ba5534f8061a582103a07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.586ex; height:2.843ex;" alt="{\displaystyle \omega (\lambda x+\mu y)=\lambda \omega (x)+\mu \omega (y),\quad x,y\in V,\quad \lambda ,\mu \in K}"></span></dd></dl> <p>Si <i>V</i> és un espai vectorial, les formes lineals de <i>V</i> se solen anomenar també <b><a href="/wiki/Covectors" class="mw-redirect" title="Covectors">covectors</a></b>, en contraposició al nom de "vectors" que hom fa servir per als elements de <i>V</i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Notació"><span id="Notaci.C3.B3"></span>Notació</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Forma_lineal&action=edit&section=1" title="Modifica la secció: Notació"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> és una forma lineal i <i>x</i> un element de <i>V</i>, de vegades s'empra la notació <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,\omega \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>ω</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,\omega \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06d73b4dc58e2e5d708cb4fb1abfc206e68d8fb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.619ex; height:2.843ex;" alt="{\displaystyle \langle x,\omega \rangle }"></span> per expressar el valor de la forma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> en l'element <i>x</i>, és a dir, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,\omega \rangle =\omega (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>ω</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,\omega \rangle =\omega (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be40baf994a0d5d664cc396e32859d5cea3f6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.302ex; height:2.843ex;" alt="{\displaystyle \langle x,\omega \rangle =\omega (x)}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Objectes_duals">Objectes duals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Forma_lineal&action=edit&section=2" title="Modifica la secció: Objectes duals"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El conjunt <span class="mwe-math-element"><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^{\ast }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{\ast }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0cb20ffe2edba87ed58c5a693fc9e8366fed329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.358ex; height:2.343ex;" alt="{\displaystyle V^{\ast }\,}"></span> de les formes lineals de l'objecte <i>V</i> a l'objecte <i>K</i> és l'<a href="/wiki/Estructura_lineal_dual" title="Estructura lineal dual">estructura lineal dual</a> de <i>V</i>. Si <i>V</i> és un <i>K</i>-mòdul o un <i>K</i>-espai vectorial, llavors <span class="mwe-math-element"><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^{\ast }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{\ast }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0cb20ffe2edba87ed58c5a693fc9e8366fed329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.358ex; height:2.343ex;" alt="{\displaystyle V^{\ast }\,}"></span> és, respectivament, el <i>K</i>-<a href="/wiki/M%C3%B2dul_dual" class="mw-redirect" title="Mòdul dual">mòdul dual</a> o l'<a href="/wiki/Espai_vectorial_dual" class="mw-redirect" title="Espai vectorial dual">espai vectorial dual</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Càlcul"><span id="C.C3.A0lcul"></span>Càlcul</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Forma_lineal&action=edit&section=3" title="Modifica la secció: Càlcul"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Com que, en tots els casos, una forma lineal no és més que un <a href="/wiki/Homomorfisme" class="mw-redirect" title="Homomorfisme">homomorfisme</a> de <i>V</i> a <i>K</i>, si <i>V</i> és un <a href="/wiki/M%C3%B2dul_lliure" title="Mòdul lliure">mòdul lliure</a> <a href="/w/index.php?title=Finitament_generat&action=edit&redlink=1" class="new" title="Finitament generat (encara no existeix)">finitament generat</a> o un <a href="/wiki/Espai_vectorial" title="Espai vectorial">espai vectorial</a> de <a href="/wiki/Dimensi%C3%B3" title="Dimensió">dimensió</a> finita, hom pot condensar tota la informació sobre una certa forma lineal <i>ω</i> en la <a href="/wiki/Matriu_(matem%C3%A0tiques)" title="Matriu (matemàtiques)">matriu</a> d'aquesta <a href="/wiki/Aplicaci%C3%B3_lineal" title="Aplicació lineal">aplicació lineal</a>. Si <i>g</i><sub>1</sub>, <i>g</i>₂, ..., <i>g</i><sub>n</sub> són els vectors d'una <a href="/wiki/Base_(%C3%A0lgebra)" title="Base (àlgebra)">base</a> de <i>V</i> i prenem { 1<sub><i>K</i></sub> } com a base de <i>K</i>, la matriu de la forma lineal <i>ω</i> és </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 \omega \quad \longleftrightarrow \quad {\begin{pmatrix}\omega (g_{1})&\omega (g_{2})&\ldots &\omega (g_{n})\end{pmatrix}}={\begin{pmatrix}\langle g_{1},\omega \rangle &\langle g_{2},\omega \rangle &\ldots &\langle g_{n},\omega \rangle \end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mspace width="1em" /> <mo stretchy="false">⟷</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </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 fence="false" stretchy="false">⟨</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>ω</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mi>ω</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mi>ω</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \quad \longleftrightarrow \quad {\begin{pmatrix}\omega (g_{1})&\omega (g_{2})&\ldots &\omega (g_{n})\end{pmatrix}}={\begin{pmatrix}\langle g_{1},\omega \rangle &\langle g_{2},\omega \rangle &\ldots &\langle g_{n},\omega \rangle \end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed62192ac565cc334049e14af9c1fb55b7caeb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:75.243ex; height:2.843ex;" alt="{\displaystyle \omega \quad \longleftrightarrow \quad {\begin{pmatrix}\omega (g_{1})&\omega (g_{2})&\ldots &\omega (g_{n})\end{pmatrix}}={\begin{pmatrix}\langle g_{1},\omega \rangle &\langle g_{2},\omega \rangle &\ldots &\langle g_{n},\omega \rangle \end{pmatrix}}}"></span></dd></dl> <p>d'una fila i <i>n</i> columnes. Per aquest motiu, les formes lineals a espais vectorials també se solen anomenar <b>vectors fila</b> en contraposició als elements de l'espai que són els <b>vectors columna</b>. </p><p>El càlcul del valor de la forma <i>ω</i> en l'element de <i>V</i> donat per les coordenades <i>v</i> = <i>λ</i><sub>1</sub><i>g</i><sub>1</sub> + <i>λ</i>₂<i>g</i>₂ + ... + <i>λ</i><sub><i>n</i></sub><i>g</i><sub><i>n</i></sub> es fa amb el producte habitual de matrius: </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 \omega (v)=\langle v,\omega \rangle \ ={\begin{pmatrix}\omega (g_{1})&\omega (g_{2})&\ldots &\omega (g_{n})\end{pmatrix}}{\begin{pmatrix}\lambda _{1}\\\lambda _{2}\\\vdots \\\lambda _{n}\end{pmatrix}}=\sum _{i=1}^{n}\langle g_{i},\omega \rangle \lambda _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo>,</mo> <mi>ω</mi> <mo fence="false" stretchy="false">⟩</mo> <mtext> </mtext> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>ω</mi> <mo fence="false" stretchy="false">⟩</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (v)=\langle v,\omega \rangle \ ={\begin{pmatrix}\omega (g_{1})&\omega (g_{2})&\ldots &\omega (g_{n})\end{pmatrix}}{\begin{pmatrix}\lambda _{1}\\\lambda _{2}\\\vdots \\\lambda _{n}\end{pmatrix}}=\sum _{i=1}^{n}\langle g_{i},\omega \rangle \lambda _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75396948b4dd8c871faee1696bd843a53a43ad86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:67.446ex; height:13.843ex;" alt="{\displaystyle \omega (v)=\langle v,\omega \rangle \ ={\begin{pmatrix}\omega (g_{1})&\omega (g_{2})&\ldots &\omega (g_{n})\end{pmatrix}}{\begin{pmatrix}\lambda _{1}\\\lambda _{2}\\\vdots \\\lambda _{n}\end{pmatrix}}=\sum _{i=1}^{n}\langle g_{i},\omega \rangle \lambda _{i}}"></span></dd></dl> <p>és a dir, es correspon amb un <a href="/wiki/Polinomi_homogeni" title="Polinomi homogeni">polinomi homogeni</a> de grau 1 en <i>n</i> variables. </p> <div class="mw-heading mw-heading2"><h2 id="Vegeu_també"><span id="Vegeu_tamb.C3.A9"></span>Vegeu també</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Forma_lineal&action=edit&section=4" title="Modifica la secció: Vegeu també"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Estructures_lineals_duals" class="mw-redirect" title="Estructures lineals duals">Estructures lineals duals</a></li> <li><a href="/wiki/Vector_fila" title="Vector fila">Vector fila</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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