Sistema d'equacions lineals - Viquipèdia, l'enciclopèdia lliure

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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-Exemple_elemental" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Exemple_elemental"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Exemple elemental</span> </div> </a> <ul id="toc-Exemple_elemental-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Forma_general" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Forma_general"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Forma general</span> </div> </a> <ul id="toc-Forma_general-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalitats" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalitats"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Generalitats</span> </div> </a> <ul id="toc-Generalitats-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Marcs_conceptuals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Marcs_conceptuals"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Marcs conceptuals</span> </div> </a> <button aria-controls="toc-Marcs_conceptuals-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>Commuta la subsecció Marcs conceptuals</span> </button> <ul id="toc-Marcs_conceptuals-sublist" class="vector-toc-list"> <li id="toc-Dependències_lineals_en_un_cert_conjunt_de_vectors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dependències_lineals_en_un_cert_conjunt_de_vectors"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Dependències lineals en un cert conjunt de vectors</span> </div> </a> <ul id="toc-Dependències_lineals_en_un_cert_conjunt_de_vectors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Antiimatge_d'un_vector_en_una_certa_aplicació_lineal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Antiimatge_d'un_vector_en_una_certa_aplicació_lineal"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Antiimatge d'un vector en una certa aplicació lineal</span> </div> </a> <ul id="toc-Antiimatge_d'un_vector_en_una_certa_aplicació_lineal-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mètodes_de_resolució" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mètodes_de_resolució"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Mètodes de resolució</span> </div> </a> <button aria-controls="toc-Mètodes_de_resolució-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>Commuta la subsecció Mètodes de resolució</span> </button> <ul id="toc-Mètodes_de_resolució-sublist" class="vector-toc-list"> <li id="toc-Resolució_pel_mètode_de_reducció_de_Gauss" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Resolució_pel_mètode_de_reducció_de_Gauss"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Resolució pel mètode de reducció de Gauss</span> </div> </a> <ul id="toc-Resolució_pel_mètode_de_reducció_de_Gauss-sublist" class="vector-toc-list"> <li id="toc-Quant_a_compatibilitat_i_determinació" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Quant_a_compatibilitat_i_determinació"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.1</span> <span>Quant a compatibilitat i determinació</span> </div> </a> <ul id="toc-Quant_a_compatibilitat_i_determinació-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Obtenció_de_les_solucions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Obtenció_de_les_solucions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.2</span> <span>Obtenció de les solucions</span> </div> </a> <ul id="toc-Obtenció_de_les_solucions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Resolució_per_la_regla_de_Cramer" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Resolució_per_la_regla_de_Cramer"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Resolució per la regla de Cramer</span> </div> </a> <ul id="toc-Resolució_per_la_regla_de_Cramer-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algorismes_alternatius" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algorismes_alternatius"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Algorismes alternatius</span> </div> </a> <ul id="toc-Algorismes_alternatius-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Sistemes_homogenis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sistemes_homogenis"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Sistemes homogenis</span> </div> </a> <ul id="toc-Sistemes_homogenis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_i_referències" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_i_referències"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes i referències</span> </div> </a> <ul id="toc-Notes_i_referències-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-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">9</span> <span>Vegeu també</span> </div> </a> <ul id="toc-Vegeu_també-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enllaços_externs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Enllaços_externs"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Enllaços externs</span> </div> </a> <ul id="toc-Enllaços_externs-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">Sistema d'equacions lineals</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 70 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-70" 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">70 llengües</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Lineares_Gleichungssystem" title="Lineares Gleichungssystem - alemany suís" lang="gsw" hreflang="gsw" data-title="Lineares Gleichungssystem" data-language-autonym="Alemannisch" data-language-local-name="alemany suís" 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/Sistema_d%27ecuacions_lineals" title="Sistema d'ecuacions lineals - aragonès" lang="an" hreflang="an" data-title="Sistema d'ecuacions lineals" data-language-autonym="Aragonés" data-language-local-name="aragonès" 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/%D9%86%D8%B8%D8%A7%D9%85_%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A7%D8%AA_%D8%AE%D8%B7%D9%8A%D8%A9" title="نظام معادلات خطية - àrab" lang="ar" hreflang="ar" data-title="نظام معادلات خطية" data-language-autonym="العربية" data-language-local-name="àrab" 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/Sistema_d%27ecuaciones_lliniales" title="Sistema d'ecuaciones lliniales - asturià" lang="ast" hreflang="ast" data-title="Sistema d'ecuaciones lliniales" data-language-autonym="Asturianu" data-language-local-name="asturià" 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_t%C9%99nlikl%C9%99r_sistemi" title="Xətti tənliklər sistemi - azerbaidjanès" lang="az" hreflang="az" data-title="Xətti tənliklər sistemi" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaidjanès" 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%D0%B8%D0%BA_%D1%82%D0%B8%D0%B3%D0%B5%D2%99%D0%BB%D3%99%D0%BC%D3%99%D0%BB%D3%99%D1%80_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0%D2%BB%D1%8B" title="Һыҙыҡлы алгебраик тигеҙләмәләр системаһы - baixkir" lang="ba" hreflang="ba" data-title="Һыҙыҡлы алгебраик тигеҙләмәләр системаһы" data-language-autonym="Башҡортса" data-language-local-name="baixkir" 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%A1%D1%96%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%BB%D1%96%D0%BD%D0%B5%D0%B9%D0%BD%D1%8B%D1%85_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D1%96%D1%87%D0%BD%D1%8B%D1%85_%D1%83%D1%80%D0%B0%D1%9E%D0%BD%D0%B5%D0%BD%D0%BD%D1%8F%D1%9E" title="Сістэма лінейных алгебраічных ураўненняў - belarús" lang="be" hreflang="be" data-title="Сістэма лінейных алгебраічных ураўненняў" data-language-autonym="Беларуская" data-language-local-name="belarús" 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%A1%D1%8B%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%BB%D1%96%D0%BD%D0%B5%D0%B9%D0%BD%D1%8B%D1%85_%D0%B0%D0%BB%D1%8C%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D1%96%D1%87%D0%BD%D1%8B%D1%85_%D1%80%D0%B0%D1%9E%D0%BD%D0%B0%D0%BD%D1%8C%D0%BD%D1%8F%D1%9E" title="Сыстэма лінейных альгебраічных раўнаньняў - Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Сыстэма лінейных альгебраічных раўнаньняў" 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%A1%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BB%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%B8_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D1%8F" title="Система линейни уравнения - búlgar" lang="bg" hreflang="bg" data-title="Система линейни уравнения" data-language-autonym="Български" data-language-local-name="búlgar" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Sistem_linearnih_jedna%C4%8Dina" title="Sistem linearnih jednačina - bosnià" lang="bs" hreflang="bs" data-title="Sistem linearnih jednačina" data-language-autonym="Bosanski" data-language-local-name="bosnià" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B3%DB%8C%D8%B3%D8%AA%D9%85%DB%8C_%DA%BE%D8%A7%D9%88%DA%A9%DB%8E%D8%B4%DB%95%DB%8C_%DA%BE%DB%8E%DA%B5%DB%8C" title="سیستمی ھاوکێشەی ھێڵی - kurd central" lang="ckb" hreflang="ckb" data-title="سیستمی ھاوکێشەی ھێڵی" data-language-autonym="کوردی" data-language-local-name="kurd central" 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/Soustava_line%C3%A1rn%C3%ADch_rovnic" title="Soustava lineárních rovnic - txec" lang="cs" hreflang="cs" data-title="Soustava lineárních rovnic" data-language-autonym="Čeština" data-language-local-name="txec" 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%C4%83%D0%BB%D0%BB%D0%B0_%D1%82%D0%B0%D0%BD%D0%BB%C4%83%D1%85%D1%81%D0%B5%D0%BD_%D1%82%D1%8B%D1%82%C4%83%D0%BC%C4%95" title="Линилле алгебрăлла танлăхсен тытăмĕ - txuvaix" lang="cv" hreflang="cv" data-title="Линилле алгебрăлла танлăхсен тытăмĕ" data-language-autonym="Чӑвашла" data-language-local-name="txuvaix" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Lineares_Gleichungssystem" title="Lineares Gleichungssystem - alemany" lang="de" hreflang="de" data-title="Lineares Gleichungssystem" data-language-autonym="Deutsch" data-language-local-name="alemany" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%8D%CF%83%CF%84%CE%B7%CE%BC%CE%B1_%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%B9%CE%BA%CF%8E%CE%BD_%CE%B5%CE%BE%CE%B9%CF%83%CF%8E%CF%83%CE%B5%CF%89%CE%BD" title="Σύστημα γραμμικών εξισώσεων - grec" lang="el" hreflang="el" data-title="Σύστημα γραμμικών εξισώσεων" data-language-autonym="Ελληνικά" data-language-local-name="grec" 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/System_of_linear_equations" title="System of linear equations - anglès" lang="en" hreflang="en" data-title="System of linear equations" data-language-autonym="English" data-language-local-name="anglès" 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/Sistemo_de_linearaj_ekvacioj" title="Sistemo de linearaj ekvacioj - esperanto" lang="eo" hreflang="eo" data-title="Sistemo de linearaj ekvacioj" 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/Sistema_de_ecuaciones_lineales" title="Sistema de ecuaciones lineales - espanyol" lang="es" hreflang="es" data-title="Sistema de ecuaciones lineales" data-language-autonym="Español" data-language-local-name="espanyol" 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/Lineaarv%C3%B5rrandis%C3%BCsteem" title="Lineaarvõrrandisüsteem - estonià" lang="et" hreflang="et" data-title="Lineaarvõrrandisüsteem" data-language-autonym="Eesti" data-language-local-name="estonià" 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/Ekuazio_linealetako_sistema" title="Ekuazio linealetako sistema - basc" lang="eu" hreflang="eu" data-title="Ekuazio linealetako sistema" data-language-autonym="Euskara" data-language-local-name="basc" 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%AF%D8%B3%D8%AA%DA%AF%D8%A7%D9%87_%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A7%D8%AA_%D8%AE%D8%B7%DB%8C" title="دستگاه معادلات خطی - persa" lang="fa" hreflang="fa" data-title="دستگاه معادلات خطی" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Lineaarinen_yht%C3%A4l%C3%B6ryhm%C3%A4" title="Lineaarinen yhtälöryhmä - finès" lang="fi" hreflang="fi" data-title="Lineaarinen yhtälöryhmä" data-language-autonym="Suomi" data-language-local-name="finès" 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/Syst%C3%A8me_d%27%C3%A9quations_lin%C3%A9aires" title="Système d'équations linéaires - francès" lang="fr" hreflang="fr" data-title="Système d'équations linéaires" data-language-autonym="Français" data-language-local-name="francès" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Sistema_de_ecuaci%C3%B3ns_lineais" title="Sistema de ecuacións lineais - gallec" lang="gl" hreflang="gl" data-title="Sistema de ecuacións lineais" data-language-autonym="Galego" data-language-local-name="gallec" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A2%D7%A8%D7%9B%D7%AA_%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%95%D7%AA_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%95%D7%AA" title="מערכת משוואות ליניאריות - hebreu" lang="he" hreflang="he" data-title="מערכת משוואות ליניאריות" data-language-autonym="עברית" data-language-local-name="hebreu" 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%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3_%E0%A4%A8%E0%A4%BF%E0%A4%95%E0%A4%BE%E0%A4%AF" title="रैखिक समीकरण निकाय - hindi" lang="hi" hreflang="hi" data-title="रैखिक समीकरण निकाय" 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/System_of_linear_equations" title="System of linear equations - hindi de Fiji" lang="hif" hreflang="hif" data-title="System of linear equations" data-language-autonym="Fiji Hindi" data-language-local-name="hindi de Fiji" 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/Sustav_linearnih_jednad%C5%BEbi" title="Sustav linearnih jednadžbi - croat" lang="hr" hreflang="hr" data-title="Sustav linearnih jednadžbi" data-language-autonym="Hrvatski" data-language-local-name="croat" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Line%C3%A1ris_egyenletrendszer" title="Lineáris egyenletrendszer - hongarès" lang="hu" hreflang="hu" data-title="Lineáris egyenletrendszer" data-language-autonym="Magyar" data-language-local-name="hongarès" 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%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%B4%D5%B6%D5%A5%D6%80%D5%AB_%D5%B0%D5%A1%D5%B4%D5%A1%D5%AF%D5%A1%D6%80%D5%A3" title="Գծային հավասարումների համակարգ - armeni" lang="hy" hreflang="hy" data-title="Գծային հավասարումների համակարգ" data-language-autonym="Հայերեն" data-language-local-name="armeni" 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/Systema_de_equationes_linear" title="Systema de equationes linear - interlingua" lang="ia" hreflang="ia" data-title="Systema de equationes 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/Sistem_persamaan_linear" title="Sistem persamaan linear - indonesi" lang="id" hreflang="id" data-title="Sistem persamaan linear" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesi" 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%ADnulegt_j%C3%B6fnuhneppi" title="Línulegt jöfnuhneppi - islandès" lang="is" hreflang="is" data-title="Línulegt jöfnuhneppi" data-language-autonym="Íslenska" data-language-local-name="islandès" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sistema_di_equazioni_lineari" title="Sistema di equazioni lineari - italià" lang="it" hreflang="it" data-title="Sistema di equazioni lineari" 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%96%B9%E7%A8%8B%E5%BC%8F%E7%B3%BB" title="線型方程式系 - japonès" lang="ja" hreflang="ja" data-title="線型方程式系" 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-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%97%B0%EB%A6%BD_%EC%9D%BC%EC%B0%A8_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="연립 일차 방정식 - coreà" lang="ko" hreflang="ko" data-title="연립 일차 방정식" data-language-autonym="한국어" data-language-local-name="coreà" 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/Systema_aequationum_linearium" title="Systema aequationum linearium - llatí" lang="la" hreflang="la" data-title="Systema aequationum linearium" data-language-autonym="Latina" data-language-local-name="llatí" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Sistema_de_equazion_linear" title="Sistema de equazion linear - llombard" lang="lmo" hreflang="lmo" data-title="Sistema de equazion linear" data-language-autonym="Lombard" data-language-local-name="llombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Line%C4%81ru_vien%C4%81dojumu_sist%C4%93ma" title="Lineāru vienādojumu sistēma - letó" lang="lv" hreflang="lv" data-title="Lineāru vienādojumu sistēma" data-language-autonym="Latviešu" data-language-local-name="letó" 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%A1%D0%B8%D1%81%D1%82%D0%B5%D0%BC_%D0%BD%D0%B0_%D0%BB%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%B8_%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D0%BA%D0%B8" title="Систем на линеарни равенки - macedoni" lang="mk" hreflang="mk" data-title="Систем на линеарни равенки" data-language-autonym="Македонски" data-language-local-name="macedoni" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sistem_persamaan_linear" title="Sistem persamaan linear - malai" lang="ms" hreflang="ms" data-title="Sistem persamaan linear" data-language-autonym="Bahasa Melayu" data-language-local-name="malai" 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/Stelsel_van_lineaire_vergelijkingen" title="Stelsel van lineaire vergelijkingen - neerlandès" lang="nl" hreflang="nl" data-title="Stelsel van lineaire vergelijkingen" data-language-autonym="Nederlands" data-language-local-name="neerlandès" 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%A6rt_likningssystem" title="Lineært likningssystem - noruec nynorsk" lang="nn" hreflang="nn" data-title="Lineært likningssystem" data-language-autonym="Norsk nynorsk" data-language-local-name="noruec 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%A6rt_ligningssystem" title="Lineært ligningssystem - noruec bokmål" lang="nb" hreflang="nb" data-title="Lineært ligningssystem" 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Sist%C3%A8ma_d%27equacions_linearas" title="Sistèma d'equacions linearas - occità" lang="oc" hreflang="oc" data-title="Sistèma d'equacions linearas" data-language-autonym="Occitan" data-language-local-name="occità" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B0%E0%A9%87%E0%A8%96%E0%A9%80_%E0%A8%B8%E0%A8%AE%E0%A9%80%E0%A8%95%E0%A8%B0%E0%A8%A8%E0%A8%BE%E0%A8%82_%E0%A8%A6%E0%A8%BE_%E0%A8%A4%E0%A9%B0%E0%A8%A4%E0%A8%B0" title="ਰੇਖੀ ਸਮੀਕਰਨਾਂ ਦਾ ਤੰਤਰ - panjabi" lang="pa" hreflang="pa" data-title="ਰੇਖੀ ਸਮੀਕਰਨਾਂ ਦਾ ਤੰਤਰ" 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/Uk%C5%82ad_r%C3%B3wna%C5%84_liniowych" title="Układ równań liniowych - polonès" lang="pl" hreflang="pl" data-title="Układ równań liniowych" data-language-autonym="Polski" data-language-local-name="polonès" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%84%DB%8C%D9%86%DB%8C%D8%B1_%D8%A7%DB%8C%DA%A9%D9%88%D8%A7%DB%8C%D8%B4%D9%86%D8%B2_%D8%AF%D8%A7_%D9%BE%D8%B1%D8%A8%D9%86%D8%AF%DA%BE" title="لینیر ایکوایشنز دا پربندھ - Western Punjabi" lang="pnb" hreflang="pnb" data-title="لینیر ایکوایشنز دا پربندھ" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Sistema_de_equa%C3%A7%C3%B5es_lineares" title="Sistema de equações lineares - portuguès" lang="pt" hreflang="pt" data-title="Sistema de equações lineares" data-language-autonym="Português" data-language-local-name="portuguès" 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/Sistem_de_ecua%C8%9Bii_liniare" title="Sistem de ecuații liniare - romanès" lang="ro" hreflang="ro" data-title="Sistem de ecuații liniare" data-language-autonym="Română" data-language-local-name="romanès" 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%A1%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BB%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D1%8B%D1%85_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D1%85_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B9" title="Система линейных алгебраических уравнений - rus" lang="ru" hreflang="ru" data-title="Система линейных алгебраических уравнений" data-language-autonym="Русский" data-language-local-name="rus" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D8%B3%D9%90%DA%8C%D9%90%D8%B1_%D9%85%D8%B3%D8%A7%D9%88%D8%A7%D8%AA%D9%8F%D9%86_%D8%AC%D9%88_%D8%B3%D8%B1%D8%B4%D8%AA%D9%88" title="سِڌِر مساواتُن جو سرشتو - sindi" lang="sd" hreflang="sd" data-title="سِڌِر مساواتُن جو سرشتو" data-language-autonym="سنڌي" data-language-local-name="sindi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Sistem_linearnih_jedna%C4%8Dina" title="Sistem linearnih jednačina - serbocroat" lang="sh" hreflang="sh" data-title="Sistem linearnih jednačina" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroat" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%92%E0%B6%9A%E0%B6%A2_%E0%B7%83%E0%B6%B8%E0%B7%93%E0%B6%9A%E0%B6%BB%E0%B6%AB_%E0%B6%B4%E0%B6%AF%E0%B7%8A%E0%B6%B0%E0%B6%AD%E0%B7%92%E0%B6%BA" title="ඒකජ සමීකරණ පද්ධතිය - singalès" lang="si" hreflang="si" data-title="ඒකජ සමීකරණ පද්ධතිය" data-language-autonym="සිංහල" data-language-local-name="singalès" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/System_of_linear_equations" title="System of linear equations - Simple English" lang="en-simple" hreflang="en-simple" data-title="System of linear equations" 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/S%C3%BAstava_line%C3%A1rnych_rovn%C3%ADc" title="Sústava lineárnych rovníc - eslovac" lang="sk" hreflang="sk" data-title="Sústava lineárnych rovníc" data-language-autonym="Slovenčina" data-language-local-name="eslovac" 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/Sistem_linearnih_ena%C4%8Db" title="Sistem linearnih enačb - eslovè" lang="sl" hreflang="sl" data-title="Sistem linearnih enačb" data-language-autonym="Slovenščina" data-language-local-name="eslovè" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D0%B8%D1%81%D1%82%D0%B5%D0%BC_%D0%BB%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%B8%D1%85_%D1%98%D0%B5%D0%B4%D0%BD%D0%B0%D1%87%D0%B8%D0%BD%D0%B0" title="Систем линеарних једначина - serbi" lang="sr" hreflang="sr" data-title="Систем линеарних једначина" data-language-autonym="Српски / srpski" data-language-local-name="serbi" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Linj%C3%A4rt_ekvationssystem" title="Linjärt ekvationssystem - suec" lang="sv" hreflang="sv" data-title="Linjärt ekvationssystem" data-language-autonym="Svenska" data-language-local-name="suec" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AF%87%E0%AE%B0%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%9A%E0%AE%AE%E0%AE%A9%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81%E0%AE%95%E0%AE%B3%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%A4%E0%AF%8A%E0%AE%95%E0%AF%81%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81" title="நேரியல் சமன்பாடுகளின் தொகுப்பு - tàmil" lang="ta" hreflang="ta" data-title="நேரியல் சமன்பாடுகளின் தொகுப்பு" data-language-autonym="தமிழ்" data-language-local-name="tàmil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B0%E0%B8%9A%E0%B8%9A%E0%B8%AA%E0%B8%A1%E0%B8%81%E0%B8%B2%E0%B8%A3%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="ระบบสมการเชิงเส้น - tai" lang="th" hreflang="th" data-title="ระบบสมการเชิงเส้น" data-language-autonym="ไทย" data-language-local-name="tai" 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//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Segell_1000_unificat_Viquip%C3%A8dia.svg/60px-Segell_1000_unificat_Viquip%C3%A8dia.svg.png 2x" data-file-width="2408" data-file-height="2896" /></a></span></div></div> </div> <div id="siteSub" class="noprint">De la Viquipèdia, l'enciclopèdia lliure</div> </div> <div id="contentSub"><div 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"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer: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>Cada equació d'un sistema d'equacions amb tres variables determina un <a href="/wiki/Pla" title="Pla">pla</a>. Resoldre el sistema és trobar els punt d'intersecció de tots els plans. En el sistema representat de la il·lustració determina tres plans (tres equacions) que es tallen en un punt, de manera que el sistema té una única solució (sistema compatible determinat).</figcaption></figure> <p>En <a href="/wiki/Matem%C3%A0tiques" title="Matemàtiques">matemàtiques</a>, un <b>sistema d'equacions lineals</b> és un conjunt d'<a href="/wiki/Equaci%C3%B3_lineal" title="Equació lineal">equacions lineals</a> que comparteixen el mateix conjunt de <a href="/wiki/Variable_(matem%C3%A0tiques)" title="Variable (matemàtiques)">variables</a> o incògnites. Per exemple: </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{alignedat}{7}3x&&\;+\;&&2y&&\;-\;&&z&&\;=\;&&1&\\2x&&\;-\;&&2y&&\;+\;&&4z&&\;=\;&&-2&\\-x&&\;+\;&&{\tfrac {1}{2}}y&&\;-\;&&z&&\;=\;&&0&\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mn>3</mn> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>2</mn> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>−</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mi>z</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd /> </mtr> <mtr> <mtd> <mn>2</mn> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>−</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>2</mn> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>4</mn> <mi>z</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd /> </mtr> <mtr> <mtd> <mo>−</mo> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>−</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mi>z</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>0</mn> </mtd> <mtd /> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{7}3x&&\;+\;&&2y&&\;-\;&&z&&\;=\;&&1&\\2x&&\;-\;&&2y&&\;+\;&&4z&&\;=\;&&-2&\\-x&&\;+\;&&{\tfrac {1}{2}}y&&\;-\;&&z&&\;=\;&&0&\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d691839a2b284331b58b0820654d32e101e26a03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:21.219ex; height:9.676ex;" alt="{\displaystyle {\begin{alignedat}{7}3x&&\;+\;&&2y&&\;-\;&&z&&\;=\;&&1&\\2x&&\;-\;&&2y&&\;+\;&&4z&&\;=\;&&-2&\\-x&&\;+\;&&{\tfrac {1}{2}}y&&\;-\;&&z&&\;=\;&&0&\end{alignedat}}}"></span></dd></dl> <p>és un sistema de tres equacions amb tres variables <span class="mwe-math-element"><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>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>. Una <b>solució</b> per a un sistema d'equacions lineals és l'assignació de valors a les variables de tal manera que els valors siguin vàlids per a totes les equacions alhora. Una solució per al sistema anterior seria: </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{alignedat}{2}x&=&1\\y&=&-2\\z&=&-2\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}x&=&1\\y&=&-2\\z&=&-2\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a8671bcb7ec1069cc0800ecb9af0ae89d18e178" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:7.505ex; height:8.843ex;" alt="{\displaystyle {\begin{alignedat}{2}x&=&1\\y&=&-2\\z&=&-2\end{alignedat}}}"></span></dd></dl> <p>que és vàlida per a les tres equacions.<sup id="cite_ref-Introducció_1-0" class="reference"><a href="#cite_note-Introducció-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Un sistema d'equacions pot tenir una única solució, diverses solucions, o cap. En funció de les possibles solucions hom parla de: </p> <ul><li>Sistema compatible: Si té solució. <ul><li>Sistema determinat: Si només té una solució.</li> <li>Sistema indeterminat: Si té un nombre infinit de solucions.</li></ul></li> <li>Sistema incompatible: Si no té cap de solució.</li></ul> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Exemple_elemental">Exemple elemental</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sistema_d%27equacions_lineals&action=edit&section=1" title="Modifica la secció: Exemple elemental"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El tipus més senzill de sistema lineal consta de dues equacions i dues variables o incògnites: </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{alignedat}{5}x&&\;+\;&&y&&\;=\;&&5&\\x&&\;-\;&&y&&\;=\;&&1&\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>5</mn> </mtd> <mtd /> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>−</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd /> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{5}x&&\;+\;&&y&&\;=\;&&5&\\x&&\;-\;&&y&&\;=\;&&1&\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd4720be39211d428ee4ae9850c129d894a20c24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.596ex; height:5.843ex;" alt="{\displaystyle {\begin{alignedat}{5}x&&\;+\;&&y&&\;=\;&&5&\\x&&\;-\;&&y&&\;=\;&&1&\end{alignedat}}}"></span></dd></dl> <p>Un mètode per a la solució d'aquest sistema és el següent. En primer lloc, a l'equació de dalt aïllarem la variable o incògnita <span class="mwe-math-element"><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>, expressant-la en termes de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></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 {x}={5}-{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x}={5}-{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d117b1a2369c5df804ed8e1f16284d399861df2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.586ex; height:2.509ex;" alt="{\displaystyle {x}={5}-{y}}"></span></dd></dl> <p>Ara substituirem a l'equació de sota la <span class="mwe-math-element"><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> per aquesta expressió: </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 \left(5-y\right)-y=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mo>−</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(5-y\right)-y=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5479de066ab67ce54fb1d520a40f7136ffca47b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.224ex; height:2.843ex;" alt="{\displaystyle \left(5-y\right)-y=1}"></span></dd></dl> <p>Això dona com a resultat una equació a la que només hi ha la variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>. Si agrupem les <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> podem escriure l'expressió anterior com: </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 5-1=2y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>2</mn> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5-1=2y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a315e5e62bf1c8d97d5197edb6daab601cd9c3bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.582ex; height:2.509ex;" alt="{\displaystyle 5-1=2y}"></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 {\frac {4}{2}}=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{2}}=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f43417c476aba12ae697d00256f9e784c3546b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.253ex; height:5.176ex;" alt="{\displaystyle {\frac {4}{2}}=y}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d401002d0346efb478da4418793e2c6174fb22c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle 2=y}"></span></dd></dl> <p>Ara que ja sabem que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/099cab48854e883602b8f188d4972969d2cf567b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=2}"></span>, podem substituir aquest valor a l'equació <span class="mwe-math-element"><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=5-y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>5</mn> <mo>−</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=5-y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33e05a16263eedb828d8d5a2ad0f9026ebc7a789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.586ex; height:2.509ex;" alt="{\displaystyle x=5-y}"></span> i el resultat serà </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=5-y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>5</mn> <mo>−</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=5-y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33e05a16263eedb828d8d5a2ad0f9026ebc7a789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.586ex; height:2.509ex;" alt="{\displaystyle x=5-y}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=5-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>5</mn> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=5-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c31dcb586355e4be502c9d9cda7e607a3b679aeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.593ex; height:2.343ex;" alt="{\displaystyle x=5-2}"></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 x=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/871a5063af170fa536b144fbcc5745146a42cc13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=3}"></span></dd></dl> <p>Aquest mètode, anomenat de substitució, es pot generalitzar per resoldre sistemes amb més de dues variables o incògnites. </p> <div class="mw-heading mw-heading2"><h2 id="Forma_general">Forma general</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sistema_d%27equacions_lineals&action=edit&section=2" title="Modifica la secció: Forma general"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De manera general, un sistema de <i>n</i> equacions lineals amb <i>m</i> incògnites es pot escriure com segueix (tenint en compte que <i>i</i> i <i>j</i> representen índexs i no <a href="/wiki/Pot%C3%A8ncia_aritm%C3%A8tica" class="mw-redirect" title="Potència aritmètica">potències</a> en expressions com <span class="mwe-math-element"><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_{i}^{j},x^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}^{j},x^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb059afca3a7a5772d12c0c412876bf15136b07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.403ex; height:3.509ex;" alt="{\displaystyle x_{i}^{j},x^{i}}"></span>): </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}\alpha _{1}^{1}x^{1}+\alpha _{2}^{1}x^{2}+\cdots +\alpha _{m}^{1}x^{m}=\beta ^{1}\\\alpha _{1}^{2}x^{1}+\alpha _{2}^{2}x^{2}+\cdots +\alpha _{m}^{2}x^{m}=\beta ^{2}\\\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\\alpha _{1}^{n}x^{1}+\alpha _{2}^{n}x^{2}+\cdots +\alpha _{m}^{n}x^{m}=\beta ^{n}\\\end{cases}}\qquad \qquad \qquad (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\alpha _{1}^{1}x^{1}+\alpha _{2}^{1}x^{2}+\cdots +\alpha _{m}^{1}x^{m}=\beta ^{1}\\\alpha _{1}^{2}x^{1}+\alpha _{2}^{2}x^{2}+\cdots +\alpha _{m}^{2}x^{m}=\beta ^{2}\\\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\\alpha _{1}^{n}x^{1}+\alpha _{2}^{n}x^{2}+\cdots +\alpha _{m}^{n}x^{m}=\beta ^{n}\\\end{cases}}\qquad \qquad \qquad (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e83bdce466b48759c3a67b7de5b11a46c3bb0cd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:52.97ex; height:12.176ex;" alt="{\displaystyle {\begin{cases}\alpha _{1}^{1}x^{1}+\alpha _{2}^{1}x^{2}+\cdots +\alpha _{m}^{1}x^{m}=\beta ^{1}\\\alpha _{1}^{2}x^{1}+\alpha _{2}^{2}x^{2}+\cdots +\alpha _{m}^{2}x^{m}=\beta ^{2}\\\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\\alpha _{1}^{n}x^{1}+\alpha _{2}^{n}x^{2}+\cdots +\alpha _{m}^{n}x^{m}=\beta ^{n}\\\end{cases}}\qquad \qquad \qquad (1)}"></span> </p> </td></tr> </tbody></table> <p>On: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{1},x^{2},...,\ldots ,x^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{1},x^{2},...,\ldots ,x^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cd0e0dc2b64b691736bc8faa6ffc10b8e152591" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.12ex; height:3.009ex;" alt="{\displaystyle x^{1},x^{2},...,\ldots ,x^{m}}"></span> són les incògnites,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{1}^{1},\alpha _{1}^{2},...,\alpha _{1}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{1}^{1},\alpha _{1}^{2},...,\alpha _{1}^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de49108222f9bc8e3b592191ca993844d66d7427" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.993ex; height:3.176ex;" alt="{\displaystyle \alpha _{1}^{1},\alpha _{1}^{2},...,\alpha _{1}^{n}}"></span> són els coeficients de les equacions del sistema, i</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ^{1},\beta ^{2},...,\beta ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{1},\beta ^{2},...,\beta ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0629311a921ce810bdf0e0b91fcd09ae773c8ede" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.542ex; height:3.009ex;" alt="{\displaystyle \beta ^{1},\beta ^{2},...,\beta ^{n}}"></span> són els termes constants.</li></ul> <p>Sovint, els coeficients i les incògnites són <a href="/wiki/Nombre_real" title="Nombre real">nombres reals</a> o <a href="/wiki/Nombre_complex" title="Nombre complex">complexos</a>, però també poden ser nombres <a href="/wiki/Nombre_enter" title="Nombre enter">enters</a> i <a href="/wiki/Nombre_racional" title="Nombre racional">racionals</a>, com són els <a href="/wiki/Polinomi" title="Polinomi">polinomis</a> i els elements d'una <a href="/wiki/Estructura_algebraica" title="Estructura algebraica">estructura algebraica</a> abstracta. </p><p><i>Solucionar el sistema</i> consisteix a trobar tots els valors de les variables (<i>incògnites</i>) <span class="mwe-math-element"><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},x^{2},\ldots ,x^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{1},x^{2},\ldots ,x^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6401d9c582701852dbcf854e4c507d540cf2c230" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.985ex; height:3.009ex;" alt="{\displaystyle x^{1},x^{2},\ldots ,x^{m}}"></span> que satisfan, alhora, les <span class="mwe-math-element"><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> equacions simultàniament. </p> <div class="mw-heading mw-heading2"><h2 id="Generalitats">Generalitats</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sistema_d%27equacions_lineals&action=edit&section=3" title="Modifica la secció: Generalitats"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La resolució de sistemes lineals d'equacions és un dels problemes més antics de les matemàtiques, els quals tenen una infinitat d'aplicacions, tant dintre de les mateixes matemàtiques com en altres ciències i tècniques, sigui el <a href="/wiki/Processament_de_senyals_digitals" title="Processament de senyals digitals">processament de senyals digitals</a>, sigui l'estimació, la predicció i, més generalment, la <a href="/wiki/Programaci%C3%B3_lineal" title="Programació lineal">programació lineal</a>, així com en l'aproximació de problemes no lineals d'<a href="/wiki/An%C3%A0lisi_num%C3%A8rica" title="Anàlisi numèrica">anàlisi numèrica</a>. Uns algorismes eficients per a resoldre sistemes d'equacions lineals són l'<a href="/wiki/M%C3%A8tode_de_reducci%C3%B3_de_Gauss" title="Mètode de reducció de Gauss">eliminació de Gauss-Jordan</a> i, millor, la <a href="/wiki/Factoritzaci%C3%B3_de_Cholesky" title="Factorització de Cholesky">factorització de Cholesky</a>.<sup id="cite_ref-Cholesky_2-0" class="reference"><a href="#cite_note-Cholesky-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Per a sistemes d'igual nombre d'equacions que d'incògnites hi ha, també, la <a href="/wiki/Regla_de_Cramer" title="Regla de Cramer">regla de Cramer</a> que, malgrat la seva importància teòrica, no és gens eficient per a sistemes amb un nombre d'incògnites superior a dos. </p> <div class="mw-heading mw-heading2"><h2 id="Marcs_conceptuals">Marcs conceptuals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sistema_d%27equacions_lineals&action=edit&section=4" title="Modifica la secció: Marcs conceptuals"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Hi ha, en principi, dos marcs conceptuals en el si dels quals podem interpretar el significat d'un cert sistema lineal d'equacions, així com el dels mètodes de resolució. Són aquests: </p> <div class="mw-heading mw-heading3"><h3 id="Dependències_lineals_en_un_cert_conjunt_de_vectors"><span id="Depend.C3.A8ncies_lineals_en_un_cert_conjunt_de_vectors"></span>Dependències lineals en un cert conjunt de vectors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sistema_d%27equacions_lineals&action=edit&section=5" title="Modifica la secció: Dependències lineals en un cert conjunt de vectors"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Podem considerar cada columna de coeficients del sistema lineal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a25115739469707c4758b189fe310a750092a80a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (1)}"></span> com a vectors d'un cert <a href="/wiki/Espai_vectorial" title="Espai vectorial">espai vectorial</a> de dimensió <span class="mwe-math-element"><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>. Aleshores tenim els <span class="mwe-math-element"><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+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f7ed29a2b4a62d3b6af05cd91a58ffc6094201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.043ex; height:2.343ex;" alt="{\displaystyle m+1}"></span> vectors </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}={\begin{pmatrix}\alpha _{1}^{1}\\\alpha _{1}^{2}\\\vdots \\\alpha _{1}^{n}\\\end{pmatrix}}\,,\quad a_{2}={\begin{pmatrix}\alpha _{2}^{1}\\\alpha _{2}^{2}\\\vdots \\\alpha _{2}^{n}\\\end{pmatrix}}\,,\quad \ldots \,,a_{m}={\begin{pmatrix}\alpha _{m}^{1}\\\alpha _{m}^{2}\\\vdots \\\alpha _{m}^{n}\\\end{pmatrix}}\,,\quad b={\begin{pmatrix}\beta ^{1}\\\beta ^{2}\\\vdots \\\beta ^{n}\\\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mo>…</mo> <mspace width="thinmathspace" /> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}={\begin{pmatrix}\alpha _{1}^{1}\\\alpha _{1}^{2}\\\vdots \\\alpha _{1}^{n}\\\end{pmatrix}}\,,\quad a_{2}={\begin{pmatrix}\alpha _{2}^{1}\\\alpha _{2}^{2}\\\vdots \\\alpha _{2}^{n}\\\end{pmatrix}}\,,\quad \ldots \,,a_{m}={\begin{pmatrix}\alpha _{m}^{1}\\\alpha _{m}^{2}\\\vdots \\\alpha _{m}^{n}\\\end{pmatrix}}\,,\quad b={\begin{pmatrix}\beta ^{1}\\\beta ^{2}\\\vdots \\\beta ^{n}\\\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ac7d70e027413667ce4eb61a762360057aff04e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:67.03ex; height:14.509ex;" alt="{\displaystyle a_{1}={\begin{pmatrix}\alpha _{1}^{1}\\\alpha _{1}^{2}\\\vdots \\\alpha _{1}^{n}\\\end{pmatrix}}\,,\quad a_{2}={\begin{pmatrix}\alpha _{2}^{1}\\\alpha _{2}^{2}\\\vdots \\\alpha _{2}^{n}\\\end{pmatrix}}\,,\quad \ldots \,,a_{m}={\begin{pmatrix}\alpha _{m}^{1}\\\alpha _{m}^{2}\\\vdots \\\alpha _{m}^{n}\\\end{pmatrix}}\,,\quad b={\begin{pmatrix}\beta ^{1}\\\beta ^{2}\\\vdots \\\beta ^{n}\\\end{pmatrix}}}"></span> </p> </td></tr> </tbody></table> <p>i el sistema <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a25115739469707c4758b189fe310a750092a80a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (1)}"></span> es pot escriure </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{1}\,{\begin{pmatrix}\alpha _{1}^{1}\\\alpha _{1}^{2}\\\vdots \\\alpha _{1}^{n}\\\end{pmatrix}}+x^{2}\,{\begin{pmatrix}\alpha _{2}^{1}\\\alpha _{2}^{2}\\\vdots \\\alpha _{2}^{n}\\\end{pmatrix}}+\cdots +x^{m}\,{\begin{pmatrix}\alpha _{m}^{1}\\\alpha _{m}^{2}\\\vdots \\\alpha _{m}^{n}\\\end{pmatrix}}={\begin{pmatrix}\beta ^{1}\\\beta ^{2}\\\vdots \\\beta ^{n}\\\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{1}\,{\begin{pmatrix}\alpha _{1}^{1}\\\alpha _{1}^{2}\\\vdots \\\alpha _{1}^{n}\\\end{pmatrix}}+x^{2}\,{\begin{pmatrix}\alpha _{2}^{1}\\\alpha _{2}^{2}\\\vdots \\\alpha _{2}^{n}\\\end{pmatrix}}+\cdots +x^{m}\,{\begin{pmatrix}\alpha _{m}^{1}\\\alpha _{m}^{2}\\\vdots \\\alpha _{m}^{n}\\\end{pmatrix}}={\begin{pmatrix}\beta ^{1}\\\beta ^{2}\\\vdots \\\beta ^{n}\\\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0f0ec55a50064cc2cfe1c398c21d310cdd2f045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:53.68ex; height:14.509ex;" alt="{\displaystyle x^{1}\,{\begin{pmatrix}\alpha _{1}^{1}\\\alpha _{1}^{2}\\\vdots \\\alpha _{1}^{n}\\\end{pmatrix}}+x^{2}\,{\begin{pmatrix}\alpha _{2}^{1}\\\alpha _{2}^{2}\\\vdots \\\alpha _{2}^{n}\\\end{pmatrix}}+\cdots +x^{m}\,{\begin{pmatrix}\alpha _{m}^{1}\\\alpha _{m}^{2}\\\vdots \\\alpha _{m}^{n}\\\end{pmatrix}}={\begin{pmatrix}\beta ^{1}\\\beta ^{2}\\\vdots \\\beta ^{n}\\\end{pmatrix}}}"></span> </p> </td></tr> </tbody></table> <p>és a dir, </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{1}a_{1}+x^{2}a_{2}+\cdots +x^{m}a_{m}=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{1}a_{1}+x^{2}a_{2}+\cdots +x^{m}a_{m}=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef61fd7c41ca15e77885808e53e280d117dc04bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.586ex; height:3.009ex;" alt="{\displaystyle x^{1}a_{1}+x^{2}a_{2}+\cdots +x^{m}a_{m}=b}"></span> </p> </td></tr> </tbody></table> <p>i, en aquest marc, solucionar el sistema lineal d'equacions consisteix a esbrinar totes les maneres possibles en les quals el vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> és combinació lineal dels vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},\ldots ,a_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},\ldots ,a_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d505725b08999ef19507fd07c8a2622661f406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.685ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},\ldots ,a_{m}}"></span>. Si el vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> no n'és combinació lineal, el sistema no té solució i es diu que és <i>incompatible</i>. Si vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> sí que ho és, el sistema té solucions i es diu <i>compatible</i> i si, a més, els vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},\ldots ,a_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},\ldots ,a_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d505725b08999ef19507fd07c8a2622661f406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.685ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},\ldots ,a_{m}}"></span> són linealment independents, l'expressió de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> com a combinació lineal de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},\ldots ,a_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},\ldots ,a_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d505725b08999ef19507fd07c8a2622661f406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.685ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},\ldots ,a_{m}}"></span> és única i el sistema té solució única: és un <i>sistema compatible determinat</i>. En cas contrari, hi ha més d'una solució i el sistema es diu <i>compatible indeterminat</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Antiimatge_d'un_vector_en_una_certa_aplicació_lineal"><span id="Antiimatge_d.27un_vector_en_una_certa_aplicaci.C3.B3_lineal"></span><a href="/wiki/Antiimatge" class="mw-redirect" title="Antiimatge">Antiimatge</a> d'un vector en una certa aplicació lineal</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sistema_d%27equacions_lineals&action=edit&section=6" title="Modifica la secció: Antiimatge d'un vector en una certa aplicació lineal"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El sistema <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a25115739469707c4758b189fe310a750092a80a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (1)}"></span> també equival a la igualtat <a href="/wiki/Matriu_(matem%C3%A0tiques)" title="Matriu (matemàtiques)">matricial</a> </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}\alpha _{1}^{1}&\alpha _{2}^{1}&\ldots &\alpha _{m}^{1}\\\alpha _{1}^{2}&\alpha _{2}^{2}&\ldots &\alpha _{m}^{2}\\\vdots &\vdots &\vdots \,\vdots \,\vdots \ &\vdots \\\alpha _{1}^{n}&\alpha _{2}^{n}&\ldots &\alpha _{m}^{n}\\\end{pmatrix}}{\begin{pmatrix}x^{1}\\x^{2}\\\vdots \\x^{m}\\\end{pmatrix}}={\begin{pmatrix}\beta ^{1}\\\beta ^{2}\\\vdots \\\beta ^{n}\\\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> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> <mspace width="thinmathspace" /> <mo>⋮</mo> <mspace width="thinmathspace" /> <mo>⋮</mo> <mtext> </mtext> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}\alpha _{1}^{1}&\alpha _{2}^{1}&\ldots &\alpha _{m}^{1}\\\alpha _{1}^{2}&\alpha _{2}^{2}&\ldots &\alpha _{m}^{2}\\\vdots &\vdots &\vdots \,\vdots \,\vdots \ &\vdots \\\alpha _{1}^{n}&\alpha _{2}^{n}&\ldots &\alpha _{m}^{n}\\\end{pmatrix}}{\begin{pmatrix}x^{1}\\x^{2}\\\vdots \\x^{m}\\\end{pmatrix}}={\begin{pmatrix}\beta ^{1}\\\beta ^{2}\\\vdots \\\beta ^{n}\\\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f42bae55f8a9109f26dc10a484aee5f9b37c2644" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:41.952ex; height:14.509ex;" alt="{\displaystyle {\begin{pmatrix}\alpha _{1}^{1}&\alpha _{2}^{1}&\ldots &\alpha _{m}^{1}\\\alpha _{1}^{2}&\alpha _{2}^{2}&\ldots &\alpha _{m}^{2}\\\vdots &\vdots &\vdots \,\vdots \,\vdots \ &\vdots \\\alpha _{1}^{n}&\alpha _{2}^{n}&\ldots &\alpha _{m}^{n}\\\end{pmatrix}}{\begin{pmatrix}x^{1}\\x^{2}\\\vdots \\x^{m}\\\end{pmatrix}}={\begin{pmatrix}\beta ^{1}\\\beta ^{2}\\\vdots \\\beta ^{n}\\\end{pmatrix}}}"></span> </p> </td></tr> </tbody></table> <p>En aquest context, la <a href="/wiki/Matriu_(matem%C3%A0tiques)" title="Matriu (matemàtiques)">matriu</a> </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{pmatrix}\alpha _{1}^{1}&\alpha _{2}^{1}&\ldots &\alpha _{m}^{1}\\\alpha _{1}^{2}&\alpha _{2}^{2}&\ldots &\alpha _{m}^{2}\\\vdots &\vdots &\vdots \,\vdots \,\vdots &\vdots \\\alpha _{1}^{n}&\alpha _{2}^{n}&\ldots &\alpha _{m}^{n}\\\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> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> <mspace width="thinmathspace" /> <mo>⋮</mo> <mspace width="thinmathspace" /> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}\alpha _{1}^{1}&\alpha _{2}^{1}&\ldots &\alpha _{m}^{1}\\\alpha _{1}^{2}&\alpha _{2}^{2}&\ldots &\alpha _{m}^{2}\\\vdots &\vdots &\vdots \,\vdots \,\vdots &\vdots \\\alpha _{1}^{n}&\alpha _{2}^{n}&\ldots &\alpha _{m}^{n}\\\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d89c6950a781bfc89d140a712a494b8c64cb41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:27.926ex; height:14.509ex;" alt="{\displaystyle A={\begin{pmatrix}\alpha _{1}^{1}&\alpha _{2}^{1}&\ldots &\alpha _{m}^{1}\\\alpha _{1}^{2}&\alpha _{2}^{2}&\ldots &\alpha _{m}^{2}\\\vdots &\vdots &\vdots \,\vdots \,\vdots &\vdots \\\alpha _{1}^{n}&\alpha _{2}^{n}&\ldots &\alpha _{m}^{n}\\\end{pmatrix}}}"></span> </p> </td></tr> </tbody></table> <p>correspon a la d'una certa <a href="/wiki/Aplicaci%C3%B3_lineal" title="Aplicació lineal">aplicació lineal</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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> d'un <a href="/wiki/Espai_vectorial" title="Espai vectorial">espai vectorial</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2d5aa67bc4c46dfb5f6a1d674998ee81063a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.39ex; height:2.509ex;" alt="{\displaystyle E_{m}}"></span> de dimensió <span class="mwe-math-element"><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> en un altre espai vectorial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6b82f2a00af6c9efd4c16d4e99329605645c0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.934ex; height:2.509ex;" alt="{\displaystyle E_{n}}"></span> de dimensió <span class="mwe-math-element"><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>: </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi :E_{m}\longleftrightarrow E_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">⟷</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi :E_{m}\longleftrightarrow E_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bca39aa58240817d882a0d1a769e975184feb55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.39ex; height:2.676ex;" alt="{\displaystyle \varphi :E_{m}\longleftrightarrow E_{n}}"></span> </p> </td></tr> </tbody></table> <p>Aleshores, si </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}_{E_{m}}=\left\{e_{1},e_{2},\ldots e_{m}\right\}\,,\quad {\mathcal {B}}_{E_{n}}=\left\{u_{1},u_{2},\ldots u_{n}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}_{E_{m}}=\left\{e_{1},e_{2},\ldots e_{m}\right\}\,,\quad {\mathcal {B}}_{E_{n}}=\left\{u_{1},u_{2},\ldots u_{n}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ce1e0402e130c6ccea26e953493fb684635addb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.929ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}_{E_{m}}=\left\{e_{1},e_{2},\ldots e_{m}\right\}\,,\quad {\mathcal {B}}_{E_{n}}=\left\{u_{1},u_{2},\ldots u_{n}\right\}}"></span> </p> </td></tr> </tbody></table> <p>són sengles bases dels espais <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2d5aa67bc4c46dfb5f6a1d674998ee81063a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.39ex; height:2.509ex;" alt="{\displaystyle E_{m}}"></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6b82f2a00af6c9efd4c16d4e99329605645c0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.934ex; height:2.509ex;" alt="{\displaystyle E_{n}}"></span>, les columnes de la matriu corresponen a les respectives imatges 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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> dels vectors de la base <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}_{E_{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}_{E_{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1245cce92c92c2fb3023d66495b570ba7e3f51d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.308ex; height:2.676ex;" alt="{\displaystyle {\mathcal {B}}_{E_{m}}}"></span> de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2d5aa67bc4c46dfb5f6a1d674998ee81063a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.39ex; height:2.509ex;" alt="{\displaystyle E_{m}}"></span> expressats en la base <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}_{E_{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}_{E_{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71c7e4e73d9fad66ae9ea926414cdf0c9879d128" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.937ex; height:2.676ex;" alt="{\displaystyle {\mathcal {B}}_{E_{n}}}"></span> de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6b82f2a00af6c9efd4c16d4e99329605645c0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.934ex; height:2.509ex;" alt="{\displaystyle E_{n}}"></span>: </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varphi \left(e_{1}\right)&=\alpha _{1}^{1}u_{1}+\alpha _{1}^{2}u_{2}+\cdots +\alpha _{1}^{n}u_{n}\\\varphi \left(e_{2}\right)&=\alpha _{2}^{1}u_{1}+\alpha _{2}^{2}u_{2}+\cdots +\alpha _{2}^{n}u_{n}\\\vdots &\vdots \\\varphi \left(e_{m}\right)&=\alpha _{m}^{1}u_{1}+\alpha _{m}^{2}u_{2}+\cdots +\alpha _{m}^{n}u_{n}\\\end{aligned}}\qquad \qquad \qquad (2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>φ</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>φ</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mi></mi> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>φ</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varphi \left(e_{1}\right)&=\alpha _{1}^{1}u_{1}+\alpha _{1}^{2}u_{2}+\cdots +\alpha _{1}^{n}u_{n}\\\varphi \left(e_{2}\right)&=\alpha _{2}^{1}u_{1}+\alpha _{2}^{2}u_{2}+\cdots +\alpha _{2}^{n}u_{n}\\\vdots &\vdots \\\varphi \left(e_{m}\right)&=\alpha _{m}^{1}u_{1}+\alpha _{m}^{2}u_{2}+\cdots +\alpha _{m}^{n}u_{n}\\\end{aligned}}\qquad \qquad \qquad (2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/300af360bf30a7588a6600fa231bb0ba51d9202d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; margin-top: -0.284ex; width:55.281ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}\varphi \left(e_{1}\right)&=\alpha _{1}^{1}u_{1}+\alpha _{1}^{2}u_{2}+\cdots +\alpha _{1}^{n}u_{n}\\\varphi \left(e_{2}\right)&=\alpha _{2}^{1}u_{1}+\alpha _{2}^{2}u_{2}+\cdots +\alpha _{2}^{n}u_{n}\\\vdots &\vdots \\\varphi \left(e_{m}\right)&=\alpha _{m}^{1}u_{1}+\alpha _{m}^{2}u_{2}+\cdots +\alpha _{m}^{n}u_{n}\\\end{aligned}}\qquad \qquad \qquad (2)}"></span> </p> </td></tr> </tbody></table> <p>la columna d'incògnites correspon a un cert vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2d5aa67bc4c46dfb5f6a1d674998ee81063a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.39ex; height:2.509ex;" alt="{\displaystyle E_{m}}"></span> expressat en la base <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}_{E_{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}_{E_{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1245cce92c92c2fb3023d66495b570ba7e3f51d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.308ex; height:2.676ex;" alt="{\displaystyle {\mathcal {B}}_{E_{m}}}"></span>: </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x^{1}e_{1}+x^{2}e_{2}+\cdots +x^{m}e_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x^{1}e_{1}+x^{2}e_{2}+\cdots +x^{m}e_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ec7f511ce4723bb4f6cb736711cc26ec4895e73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.479ex; height:3.009ex;" alt="{\displaystyle x=x^{1}e_{1}+x^{2}e_{2}+\cdots +x^{m}e_{m}}"></span> </p> </td></tr> </tbody></table> <p>i la columna de termes independents correspon a un cert vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6b82f2a00af6c9efd4c16d4e99329605645c0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.934ex; height:2.509ex;" alt="{\displaystyle E_{n}}"></span> expressat en la base <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}_{E_{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}_{E_{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71c7e4e73d9fad66ae9ea926414cdf0c9879d128" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.937ex; height:2.676ex;" alt="{\displaystyle {\mathcal {B}}_{E_{n}}}"></span>: </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=\beta ^{1}u_{1}+\beta ^{2}u_{2}+\cdots +\beta ^{n}u_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=\beta ^{1}u_{1}+\beta ^{2}u_{2}+\cdots +\beta ^{n}u_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1a817f9d719aa02d379a0e87439ffc98c662869" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.994ex; height:3.009ex;" alt="{\displaystyle b=\beta ^{1}u_{1}+\beta ^{2}u_{2}+\cdots +\beta ^{n}u_{n}}"></span> </p> </td></tr> </tbody></table> <p>i ara, en aquest altre marc, solucionar el sistema consisteix a trobar tots els vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in E_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in E_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2ba1471584efc98ec552c47d31c4f7c82056264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.561ex; height:2.509ex;" alt="{\displaystyle x\in E_{m}}"></span> pels quals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \left(x\right)=b\in E_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>b</mi> <mo>∈</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \left(x\right)=b\in E_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6830c631c78e27487d138d6a355a444cb36fd603" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.917ex; height:2.843ex;" alt="{\displaystyle \varphi \left(x\right)=b\in E_{n}}"></span>, és a dir, trobar tota la <a href="/wiki/Antiimatge" class="mw-redirect" title="Antiimatge">antiimatge</a> del vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in E_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>∈</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\in E_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0878f6aec934acb4331888d4acfc08fec96f0e5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.772ex; height:2.509ex;" alt="{\displaystyle b\in E_{n}}"></span>. </p><p>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\notin \varphi \left(E_{m}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>∉</mo> <mi>φ</mi> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\notin \varphi \left(E_{m}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3a4a4633e728433d8ee0718f0532876ca1a77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.945ex; height:2.843ex;" alt="{\displaystyle b\notin \varphi \left(E_{m}\right)}"></span>, és a dir, 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 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> no és de la imatge de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>, el vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in E_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in E_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2ba1471584efc98ec552c47d31c4f7c82056264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.561ex; height:2.509ex;" alt="{\displaystyle x\in E_{m}}"></span> no existeix pas i, aleshores, el sistema no té solució: és <i>incompatible</i>. Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in \varphi \left(E_{m}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>∈</mo> <mi>φ</mi> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\in \varphi \left(E_{m}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08654834f3c3d48559cb04a67c8a69a13d60a28c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.945ex; height:2.843ex;" alt="{\displaystyle b\in \varphi \left(E_{m}\right)}"></span>, és a dir, 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 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> pertany a la imatge de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>, hi ha vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in E_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in E_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2ba1471584efc98ec552c47d31c4f7c82056264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.561ex; height:2.509ex;" alt="{\displaystyle x\in E_{m}}"></span> que fan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (x)=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (x)=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c7afbd39efa04f7e9bdb77f4e3445fbd896757" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.755ex; height:2.843ex;" alt="{\displaystyle \varphi (x)=b}"></span> i el sistema té solució: és <i>compatible</i>. Si, a més, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> és una aplicació lineal injectiva, el vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> és únic, la solució del sistema és única i el sistema és <i>determinat</i>. Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> no és injectiva, hi ha més d'una solució i el sistema es diu <i>indeterminat</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Mètodes_de_resolució"><span id="M.C3.A8todes_de_resoluci.C3.B3"></span>Mètodes de resolució</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sistema_d%27equacions_lineals&action=edit&section=7" title="Modifica la secció: Mètodes de resolució"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Resolució_pel_mètode_de_reducció_de_Gauss"><span id="Resoluci.C3.B3_pel_m.C3.A8tode_de_reducci.C3.B3_de_Gauss"></span>Resolució pel mètode de reducció de Gauss</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sistema_d%27equacions_lineals&action=edit&section=8" title="Modifica la secció: Resolució pel mètode de reducció de Gauss"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r30997230">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">Article principal: <a href="/wiki/M%C3%A8tode_de_reducci%C3%B3_de_Gauss" title="Mètode de reducció de Gauss">Mètode de reducció de Gauss</a></div> <p>Els dos marcs conceptuals esmentats porten, tanmateix, al mateix problema. Trobar els vectors <span class="mwe-math-element"><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> que fan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (x)=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (x)=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c7afbd39efa04f7e9bdb77f4e3445fbd896757" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.755ex; height:2.843ex;" alt="{\displaystyle \varphi (x)=b}"></span> consisteix en trobar els coeficients (les incògnites!) <span class="mwe-math-element"><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^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be8cd88951c7c0e3b181f956531b0a878bbed203" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.129ex; height:2.676ex;" alt="{\displaystyle x^{i}}"></span> a </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varphi (x)&=\varphi \left(x^{1}e_{1}+x^{2}e_{2}+\cdots +x^{m}e_{m}\right)=\\&=x^{1}\varphi \left(e_{1}\right)+x^{2}\varphi \left(e_{2}\right)+\cdots +x^{m}\varphi \left(e_{m}\right)=b\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>φ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mi>φ</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>φ</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mi>φ</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varphi (x)&=\varphi \left(x^{1}e_{1}+x^{2}e_{2}+\cdots +x^{m}e_{m}\right)=\\&=x^{1}\varphi \left(e_{1}\right)+x^{2}\varphi \left(e_{2}\right)+\cdots +x^{m}\varphi \left(e_{m}\right)=b\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7cac1bf05f6f9765fa9e7e61a87d54dc107a0be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:49.806ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}\varphi (x)&=\varphi \left(x^{1}e_{1}+x^{2}e_{2}+\cdots +x^{m}e_{m}\right)=\\&=x^{1}\varphi \left(e_{1}\right)+x^{2}\varphi \left(e_{2}\right)+\cdots +x^{m}\varphi \left(e_{m}\right)=b\end{aligned}}}"></span> </p> </td></tr> </tbody></table> <p>i tornem a estar davant del problema d'esbrinar totes les maneres possibles, si n'hi ha, en les quals el vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> és combinació lineal dels vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \left(e_{1}\right),\varphi \left(e_{2}\right),\ldots ,\varphi \left(e_{m}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mi>φ</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>φ</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \left(e_{1}\right),\varphi \left(e_{2}\right),\ldots ,\varphi \left(e_{m}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30db4347ae8492269d147f0df055d7973d17c03e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.17ex; height:2.843ex;" alt="{\displaystyle \varphi \left(e_{1}\right),\varphi \left(e_{2}\right),\ldots ,\varphi \left(e_{m}\right)}"></span>, és a dir, totes les maneres possibles en les quals el vector </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b={\begin{pmatrix}\beta ^{1}\\\beta ^{2}\\\vdots \\\beta ^{n}\\\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b={\begin{pmatrix}\beta ^{1}\\\beta ^{2}\\\vdots \\\beta ^{n}\\\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08a23d15eaac5aa75facfe4c0d2b75df0da873be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:11.47ex; height:13.843ex;" alt="{\displaystyle b={\begin{pmatrix}\beta ^{1}\\\beta ^{2}\\\vdots \\\beta ^{n}\\\end{pmatrix}}}"></span> </p> </td></tr> </tbody></table> <p>és combinació lineal dels vectors </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \left(e_{1}\right)={\begin{pmatrix}\alpha _{1}^{1}\\\alpha _{1}^{2}\\\vdots \\\alpha _{1}^{n}\\\end{pmatrix}}\,,\quad \varphi \left(e_{2}\right)={\begin{pmatrix}\alpha _{2}^{1}\\\alpha _{2}^{2}\\\vdots \\\alpha _{2}^{n}\\\end{pmatrix}}\,,\quad \ldots \,,\varphi \left(e_{m}\right)={\begin{pmatrix}\alpha _{m}^{1}\\\alpha _{m}^{2}\\\vdots \\\alpha _{m}^{n}\\\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>φ</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mo>…</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mi>φ</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \left(e_{1}\right)={\begin{pmatrix}\alpha _{1}^{1}\\\alpha _{1}^{2}\\\vdots \\\alpha _{1}^{n}\\\end{pmatrix}}\,,\quad \varphi \left(e_{2}\right)={\begin{pmatrix}\alpha _{2}^{1}\\\alpha _{2}^{2}\\\vdots \\\alpha _{2}^{n}\\\end{pmatrix}}\,,\quad \ldots \,,\varphi \left(e_{m}\right)={\begin{pmatrix}\alpha _{m}^{1}\\\alpha _{m}^{2}\\\vdots \\\alpha _{m}^{n}\\\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a10b7ee81b9fcc00d703775e12a219e0b203793" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:62.527ex; height:14.509ex;" alt="{\displaystyle \varphi \left(e_{1}\right)={\begin{pmatrix}\alpha _{1}^{1}\\\alpha _{1}^{2}\\\vdots \\\alpha _{1}^{n}\\\end{pmatrix}}\,,\quad \varphi \left(e_{2}\right)={\begin{pmatrix}\alpha _{2}^{1}\\\alpha _{2}^{2}\\\vdots \\\alpha _{2}^{n}\\\end{pmatrix}}\,,\quad \ldots \,,\varphi \left(e_{m}\right)={\begin{pmatrix}\alpha _{m}^{1}\\\alpha _{m}^{2}\\\vdots \\\alpha _{m}^{n}\\\end{pmatrix}}}"></span> </p> </td></tr> </tbody></table> <p>i, si posem <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \left(e_{1}\right)=a_{1},\varphi \left(e_{2}\right)=a_{2},\ldots ,\varphi \left(e_{m}\right)=a_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>φ</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>φ</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \left(e_{1}\right)=a_{1},\varphi \left(e_{2}\right)=a_{2},\ldots ,\varphi \left(e_{m}\right)=a_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb106d4de052d460de1c5f8d16b9546d5368b49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.164ex; height:2.843ex;" alt="{\displaystyle \varphi \left(e_{1}\right)=a_{1},\varphi \left(e_{2}\right)=a_{2},\ldots ,\varphi \left(e_{m}\right)=a_{m}}"></span>, es tracta del mateix problema, exactament, plantejat al primer dels marcs conceptuals exposats. </p><p>Aquest problema té la seva resposta en el <a href="/wiki/M%C3%A8tode_de_reducci%C3%B3_de_Gauss" title="Mètode de reducció de Gauss">mètode de reducció de Gauss</a>. Es tracta de considerar les dues matrius, </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{pmatrix}\alpha _{1}^{1}&\alpha _{2}^{1}&\ldots &\alpha _{m}^{1}\\\alpha _{1}^{2}&\alpha _{2}^{2}&\ldots &\alpha _{m}^{2}\\\vdots &\vdots &\vdots \,\vdots \,\vdots &\vdots \\\alpha _{1}^{n}&\alpha _{2}^{n}&\ldots &\alpha _{m}^{n}\\\end{pmatrix}}\,,\qquad (A|b)={\begin{pmatrix}\alpha _{1}^{1}&\alpha _{2}^{1}&\ldots &\alpha _{m}^{1}&\beta ^{1}\\\alpha _{1}^{2}&\alpha _{2}^{2}&\ldots &\alpha _{m}^{2}&\beta ^{2}\\\vdots &\vdots &\vdots \,\vdots \,\vdots &\vdots &\vdots \\\alpha _{1}^{n}&\alpha _{2}^{n}&\ldots &\alpha _{m}^{n}&\beta ^{n}\\\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> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> <mspace width="thinmathspace" /> <mo>⋮</mo> <mspace width="thinmathspace" /> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> <mtd> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> <mspace width="thinmathspace" /> <mo>⋮</mo> <mspace width="thinmathspace" /> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mtd> <mtd> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}\alpha _{1}^{1}&\alpha _{2}^{1}&\ldots &\alpha _{m}^{1}\\\alpha _{1}^{2}&\alpha _{2}^{2}&\ldots &\alpha _{m}^{2}\\\vdots &\vdots &\vdots \,\vdots \,\vdots &\vdots \\\alpha _{1}^{n}&\alpha _{2}^{n}&\ldots &\alpha _{m}^{n}\\\end{pmatrix}}\,,\qquad (A|b)={\begin{pmatrix}\alpha _{1}^{1}&\alpha _{2}^{1}&\ldots &\alpha _{m}^{1}&\beta ^{1}\\\alpha _{1}^{2}&\alpha _{2}^{2}&\ldots &\alpha _{m}^{2}&\beta ^{2}\\\vdots &\vdots &\vdots \,\vdots \,\vdots &\vdots &\vdots \\\alpha _{1}^{n}&\alpha _{2}^{n}&\ldots &\alpha _{m}^{n}&\beta ^{n}\\\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac431d7f8dd12e30a07097ba32090db49805cb2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:70.249ex; height:14.509ex;" alt="{\displaystyle A={\begin{pmatrix}\alpha _{1}^{1}&\alpha _{2}^{1}&\ldots &\alpha _{m}^{1}\\\alpha _{1}^{2}&\alpha _{2}^{2}&\ldots &\alpha _{m}^{2}\\\vdots &\vdots &\vdots \,\vdots \,\vdots &\vdots \\\alpha _{1}^{n}&\alpha _{2}^{n}&\ldots &\alpha _{m}^{n}\\\end{pmatrix}}\,,\qquad (A|b)={\begin{pmatrix}\alpha _{1}^{1}&\alpha _{2}^{1}&\ldots &\alpha _{m}^{1}&\beta ^{1}\\\alpha _{1}^{2}&\alpha _{2}^{2}&\ldots &\alpha _{m}^{2}&\beta ^{2}\\\vdots &\vdots &\vdots \,\vdots \,\vdots &\vdots &\vdots \\\alpha _{1}^{n}&\alpha _{2}^{n}&\ldots &\alpha _{m}^{n}&\beta ^{n}\\\end{pmatrix}}}"></span> </p> </td></tr> </tbody></table> <p>respectivament, la <i>matriu del sistema</i> i la <i>matriu ampliada del sistema</i>, fer-ne la reducció, comparar els <a href="/wiki/Rang_d%27una_matriu" class="mw-redirect" title="Rang d'una matriu">rangs</a> de la matriu del sistema i el de la <a href="/wiki/Matriu_ampliada" title="Matriu ampliada">matriu ampliada</a>, i expressar convenientment les relacions de dependència lineal que es posaran de manifest. </p> <div class="mw-heading mw-heading4"><h4 id="Quant_a_compatibilitat_i_determinació"><span id="Quant_a_compatibilitat_i_determinaci.C3.B3"></span>Quant a compatibilitat i determinació</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sistema_d%27equacions_lineals&action=edit&section=9" title="Modifica la secció: Quant a compatibilitat i determinació"><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 {\mbox{rang}}\,A<{\mbox{rang}}\,(A|b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>rang</mtext> </mstyle> </mrow> <mspace width="thinmathspace" /> <mi>A</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>rang</mtext> </mstyle> </mrow> <mspace width="thinmathspace" /> <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 {\mbox{rang}}\,A<{\mbox{rang}}\,(A|b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26b3a00838e7a8e6fe48a5e3da4e35357aaceb87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.871ex; height:2.843ex;" alt="{\displaystyle {\mbox{rang}}\,A<{\mbox{rang}}\,(A|b)}"></span>, aleshores això indica que el vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> és independent dels vectors de la matriu <span class="mwe-math-element"><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>, en conseqüència, no pot ser-ne una combinació lineal i el sistema no té solució: es diu que és <i>incompatible</i>. En canvi, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{rang}}\,A={\mbox{rang}}\,(A|b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>rang</mtext> </mstyle> </mrow> <mspace width="thinmathspace" /> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>rang</mtext> </mstyle> </mrow> <mspace width="thinmathspace" /> <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 {\mbox{rang}}\,A={\mbox{rang}}\,(A|b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a097ab40ca34896affaf505886c051da185cb4b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.871ex; height:2.843ex;" alt="{\displaystyle {\mbox{rang}}\,A={\mbox{rang}}\,(A|b)}"></span> implica que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> no és independent dels vectors de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> i, per tant, que sí que n'és una combinació lineal i el sistema sí que te solució: és <i>compatible</i>. Si, a més, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{rang}}\,A={\mbox{rang}}\,(A|b)=m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>rang</mtext> </mstyle> </mrow> <mspace width="thinmathspace" /> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>rang</mtext> </mstyle> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{rang}}\,A={\mbox{rang}}\,(A|b)=m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23ef7cc9fb3b4217e57e655c02881d738586085a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.01ex; height:2.843ex;" alt="{\displaystyle {\mbox{rang}}\,A={\mbox{rang}}\,(A|b)=m}"></span>, els vectors de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> són linealment independents i l'expressió de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> és única: el sistema té solució única i és <i>compatible i determinat</i>. En canvi, 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 {\mbox{rang}}\,A={\mbox{rang}}\,(A|b)<m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>rang</mtext> </mstyle> </mrow> <mspace width="thinmathspace" /> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>rang</mtext> </mstyle> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{rang}}\,A={\mbox{rang}}\,(A|b)<m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6add70aeeec96a21230760994cdb9dda4898190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.01ex; height:2.843ex;" alt="{\displaystyle {\mbox{rang}}\,A={\mbox{rang}}\,(A|b)<m}"></span>, els vectors de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> no són independents i la solució no és única: el sistema és <i>compatible i indeterminat</i>. </p> <div class="mw-heading mw-heading4"><h4 id="Obtenció_de_les_solucions"><span id="Obtenci.C3.B3_de_les_solucions"></span>Obtenció de les solucions</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sistema_d%27equacions_lineals&action=edit&section=10" title="Modifica la secció: Obtenció de les solucions"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Il·lustrarem ara com s'obté la solució general d'un sistema a partir de la reducció de la <a href="/wiki/Matriu_ampliada" title="Matriu ampliada">matriu ampliada</a> mitjançant un exemple. Considerem el sistema </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\begin{aligned}5x-y+11z+6t&=3\\3x+4y+2z+2t&=8\\2x+y+3z-t&=6\\x+3y-z-2t&=7\end{aligned}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>5</mn> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>+</mo> <mn>11</mn> <mi>z</mi> <mo>+</mo> <mn>6</mn> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>4</mn> <mi>y</mi> <mo>+</mo> <mn>2</mn> <mi>z</mi> <mo>+</mo> <mn>2</mn> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mn>3</mn> <mi>z</mi> <mo>−</mo> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mn>2</mn> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>7</mn> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\begin{aligned}5x-y+11z+6t&=3\\3x+4y+2z+2t&=8\\2x+y+3z-t&=6\\x+3y-z-2t&=7\end{aligned}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d7a860344ced0bbb93c3c30a8fba1307bc4f9ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:25.414ex; height:11.843ex;" alt="{\displaystyle {\begin{cases}{\begin{aligned}5x-y+11z+6t&=3\\3x+4y+2z+2t&=8\\2x+y+3z-t&=6\\x+3y-z-2t&=7\end{aligned}}\end{cases}}}"></span> </p> </td></tr> </tbody></table> <p>de matriu ampliada </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}5&-1&11&6&3\\3&4&2&2&8\\2&1&3&-1&6\\1&3&-1&-2&7\\\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>5</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>11</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>7</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}5&-1&11&6&3\\3&4&2&2&8\\2&1&3&-1&6\\1&3&-1&-2&7\\\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/117a8914f5e2ab3329108ef689eab618e12ba35f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:25.345ex; height:12.509ex;" alt="{\displaystyle {\begin{pmatrix}5&-1&11&6&3\\3&4&2&2&8\\2&1&3&-1&6\\1&3&-1&-2&7\\\end{pmatrix}}}"></span> </p> </td></tr> </tbody></table> <p>equivalent a </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xa_{1}+ya_{2}+za_{3}+ta_{4}=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>y</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>z</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mi>t</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xa_{1}+ya_{2}+za_{3}+ta_{4}=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/076dc001bb0e5f1d4f63d6677dd1c25d3a8369cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.166ex; height:2.509ex;" alt="{\displaystyle xa_{1}+ya_{2}+za_{3}+ta_{4}=b}"></span> </p> </td></tr> </tbody></table> <p>amb </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}={\begin{pmatrix}5\\3\\2\\1\end{pmatrix}}\,,\quad a_{2}={\begin{pmatrix}-1\\4\\1\\3\end{pmatrix}}\,,\quad a_{3}={\begin{pmatrix}11\\2\\3\\-1\end{pmatrix}}\,,\quad a_{4}={\begin{pmatrix}6\\2\\-1\\-2\end{pmatrix}}\,,\quad b={\begin{pmatrix}3\\8\\6\\7\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>11</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>7</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}={\begin{pmatrix}5\\3\\2\\1\end{pmatrix}}\,,\quad a_{2}={\begin{pmatrix}-1\\4\\1\\3\end{pmatrix}}\,,\quad a_{3}={\begin{pmatrix}11\\2\\3\\-1\end{pmatrix}}\,,\quad a_{4}={\begin{pmatrix}6\\2\\-1\\-2\end{pmatrix}}\,,\quad b={\begin{pmatrix}3\\8\\6\\7\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb391a1f4cc8071b24a01bf8de128afab09047f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:75.929ex; height:12.509ex;" alt="{\displaystyle a_{1}={\begin{pmatrix}5\\3\\2\\1\end{pmatrix}}\,,\quad a_{2}={\begin{pmatrix}-1\\4\\1\\3\end{pmatrix}}\,,\quad a_{3}={\begin{pmatrix}11\\2\\3\\-1\end{pmatrix}}\,,\quad a_{4}={\begin{pmatrix}6\\2\\-1\\-2\end{pmatrix}}\,,\quad b={\begin{pmatrix}3\\8\\6\\7\end{pmatrix}}}"></span> </p> </td></tr> </tbody></table> <p>Una vegada feta la reducció de Gauss, obtenim la matriu </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}1&0&2&0&2\\0&1&-1&0&1\\0&0&0&1&-1\\0&0&0&0&0\\\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&0&2&0&2\\0&1&-1&0&1\\0&0&0&1&-1\\0&0&0&0&0\\\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e586756f85ef49d59052eb0253f3983414523124" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:23.537ex; height:12.509ex;" alt="{\displaystyle {\begin{pmatrix}1&0&2&0&2\\0&1&-1&0&1\\0&0&0&1&-1\\0&0&0&0&0\\\end{pmatrix}}}"></span> </p> </td></tr> </tbody></table> <p>El rang de la matriu del sistema és 3 i el de la matriu ampliada també és 3. Per tant el sistema és <i>compatible</i>. Però com que aquest rang, 3, és més petit que el nombre d'incògnites, que és 4, el sistema és <i>indeterminat</i>. </p><p>La relació ara òbvia: </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2a_{1}+a_{2}-a_{4}=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a_{1}+a_{2}-a_{4}=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96c9b470d5ee2c2e229e83dfad01f2ca4396e94d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.791ex; height:2.509ex;" alt="{\displaystyle 2a_{1}+a_{2}-a_{4}=b}"></span> </p> </td></tr> </tbody></table> <p>com que el problema consistia, precisament en trobar els coeficients dels vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"></span> en una combinació lineal que dona el vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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>, ens proporciona una <i>solució particular</i> del sistema: </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\begin{aligned}x&=2\\y&=1\\z&=0\\t&=-1\\\end{aligned}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\begin{aligned}x&=2\\y&=1\\z&=0\\t&=-1\\\end{aligned}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad760d909b2acf4496413ab3d7f4ea1505c818e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:10.968ex; height:11.843ex;" alt="{\displaystyle {\begin{cases}{\begin{aligned}x&=2\\y&=1\\z&=0\\t&=-1\\\end{aligned}}\end{cases}}}"></span> </p> </td></tr> </tbody></table> <p>i, a partir de la relació, també òbvia, </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{3}=2a_{1}-a_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{3}=2a_{1}-a_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d835a093d898c5cd7a2b4339d28f347af9a58eb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.953ex; height:2.509ex;" alt="{\displaystyle a_{3}=2a_{1}-a_{2}}"></span> </p> </td></tr> </tbody></table> <p>podem escriure, per 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 \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> arbitrari, </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2a_{1}+a_{2}+\lambda \left(2a_{1}-a_{2}-a_{3}\right)-a_{4}=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>λ</mi> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a_{1}+a_{2}+\lambda \left(2a_{1}-a_{2}-a_{3}\right)-a_{4}=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b477e76b08cb69b00870f38043b3751577c11976" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.879ex; height:2.843ex;" alt="{\displaystyle 2a_{1}+a_{2}+\lambda \left(2a_{1}-a_{2}-a_{3}\right)-a_{4}=b}"></span> </p> </td></tr> </tbody></table> <p>és a dir, </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2+2\lambda )a_{1}+(1-\lambda )a_{2}-\lambda a_{3}-a_{4}=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mi>λ</mi> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>λ</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2+2\lambda )a_{1}+(1-\lambda )a_{2}-\lambda a_{3}-a_{4}=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf52d254efb998ae058e1f9dd3ac7888b4e92aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.606ex; height:2.843ex;" alt="{\displaystyle (2+2\lambda )a_{1}+(1-\lambda )a_{2}-\lambda a_{3}-a_{4}=b}"></span> </p> </td></tr> </tbody></table> <p>que, per les mateixes raons, dona la <i>solució general</i> del sistema: </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\begin{aligned}x&=2+2\lambda \\y&=1-\lambda \\z&=-\lambda \\t&=-1\\\end{aligned}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mi>λ</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>λ</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>λ</mi> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\begin{aligned}x&=2+2\lambda \\y&=1-\lambda \\z&=-\lambda \\t&=-1\\\end{aligned}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0589e5a44f70d29d0f88e7dd8a7a1648e02f8576" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:14.518ex; height:11.843ex;" alt="{\displaystyle {\begin{cases}{\begin{aligned}x&=2+2\lambda \\y&=1-\lambda \\z&=-\lambda \\t&=-1\\\end{aligned}}\end{cases}}}"></span> </p> </td></tr> </tbody></table> <p>que se sol escriure </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}x\\y\\z\\t\\\end{pmatrix}}={\begin{pmatrix}2\\1\\0\\-1\\\end{pmatrix}}+\lambda {\begin{pmatrix}2\\-1\\-1\\0\\\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <mi>t</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>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}x\\y\\z\\t\\\end{pmatrix}}={\begin{pmatrix}2\\1\\0\\-1\\\end{pmatrix}}+\lambda {\begin{pmatrix}2\\-1\\-1\\0\\\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c5f23749ed411f7b99d64954caa1e00a0461e75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:29.02ex; height:12.509ex;" alt="{\displaystyle {\begin{pmatrix}x\\y\\z\\t\\\end{pmatrix}}={\begin{pmatrix}2\\1\\0\\-1\\\end{pmatrix}}+\lambda {\begin{pmatrix}2\\-1\\-1\\0\\\end{pmatrix}}}"></span> </p> </td></tr> </tbody></table> <p>Observem com els valors <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></span> de la solució particular ja apareixen com a components del vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> a la matriu reduïda, i com els valors <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></span> del vector afegit per a la solució general ja apareixen com a components del vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/602d08dd865689204f563ce6f0de095c8ca67410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{3}}"></span> també a la matriu reduïda. </p><p>Si un altre sistema de quatre equacions en les cinc incògnites <span class="mwe-math-element"><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,t,u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z,t,u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30ab7d45b41dd47dba96ff7dcae2b767066d22d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.878ex; height:2.343ex;" alt="{\displaystyle x,y,z,t,u}"></span> té, després de la reducció, com a <a href="/wiki/Matriu_ampliada" title="Matriu ampliada">matriu ampliada</a>, la següent, </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}1&3&0&0&2&7\\0&0&1&0&-5&8\\0&0&0&1&4&-6\\0&0&0&0&0&0\\\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>5</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mo>−</mo> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&3&0&0&2&7\\0&0&1&0&-5&8\\0&0&0&1&4&-6\\0&0&0&0&0&0\\\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e665afc704236f55e9df3d4e4cbb53e11c2d8fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:27.022ex; height:12.509ex;" alt="{\displaystyle {\begin{pmatrix}1&3&0&0&2&7\\0&0&1&0&-5&8\\0&0&0&1&4&-6\\0&0&0&0&0&0\\\end{pmatrix}}}"></span> </p> </td></tr> </tbody></table> <p>les relacions </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=7a_{1}+8a_{3}-6a_{4}\,,\quad a_{2}=3a_{1}\,,\quad a_{5}=2a_{1}-5a_{3}+4a_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>7</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>8</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mn>6</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>3</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>5</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mn>4</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=7a_{1}+8a_{3}-6a_{4}\,,\quad a_{2}=3a_{1}\,,\quad a_{5}=2a_{1}-5a_{3}+4a_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf137473185ec4c0ed4ae4d1380a8979be46378" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:57.835ex; height:2.509ex;" alt="{\displaystyle b=7a_{1}+8a_{3}-6a_{4}\,,\quad a_{2}=3a_{1}\,,\quad a_{5}=2a_{1}-5a_{3}+4a_{4}}"></span> </p> </td></tr> </tbody></table> <p>donen, amb <span class="mwe-math-element"><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> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> arbitraris, </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=7a_{1}+8a_{3}-6a_{4}+\lambda \left(3a_{1}-a_{2}\right)+\mu \left(2a_{1}-5a_{3}+4a_{4}-a_{5}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>7</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>8</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mn>6</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <mi>λ</mi> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>μ</mi> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>5</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mn>4</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=7a_{1}+8a_{3}-6a_{4}+\lambda \left(3a_{1}-a_{2}\right)+\mu \left(2a_{1}-5a_{3}+4a_{4}-a_{5}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e79c3d8dd4caede80f971c02dc1534f6757a6719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.663ex; height:2.843ex;" alt="{\displaystyle b=7a_{1}+8a_{3}-6a_{4}+\lambda \left(3a_{1}-a_{2}\right)+\mu \left(2a_{1}-5a_{3}+4a_{4}-a_{5}\right)}"></span> </p> </td></tr> </tbody></table> <p>és a dir, </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (7+3\lambda +2\mu )a_{1}-\lambda a_{2}+(8-5\mu )a_{3}+(-6+4\mu )a_{4}-\mu a_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>7</mn> <mo>+</mo> <mn>3</mn> <mi>λ</mi> <mo>+</mo> <mn>2</mn> <mi>μ</mi> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>λ</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mn>8</mn> <mo>−</mo> <mn>5</mn> <mi>μ</mi> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>6</mn> <mo>+</mo> <mn>4</mn> <mi>μ</mi> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (7+3\lambda +2\mu )a_{1}-\lambda a_{2}+(8-5\mu )a_{3}+(-6+4\mu )a_{4}-\mu a_{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a9f15dbc184ed055107be8e7021c80f6e6fb402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.834ex; height:2.843ex;" alt="{\displaystyle (7+3\lambda +2\mu )a_{1}-\lambda a_{2}+(8-5\mu )a_{3}+(-6+4\mu )a_{4}-\mu a_{5}}"></span> </p> </td></tr> </tbody></table> <p>i la solució general és </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\begin{aligned}x&=7+3\lambda +2\mu \\y&=-\lambda \\z&=8-5\mu \\t&=-6+4\mu \\u&=-\mu \\\end{aligned}}\end{cases}}\,,\qquad {\mbox{o}}\qquad {\begin{pmatrix}x\\y\\z\\t\\u\\\end{pmatrix}}={\begin{pmatrix}7\\0\\8\\-6\\0\\\end{pmatrix}}+\lambda {\begin{pmatrix}3\\-1\\0\\0\\0\\\end{pmatrix}}+\mu {\begin{pmatrix}2\\0\\-5\\4\\-1\\\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>7</mn> <mo>+</mo> <mn>3</mn> <mi>λ</mi> <mo>+</mo> <mn>2</mn> <mi>μ</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>λ</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>8</mn> <mo>−</mo> <mn>5</mn> <mi>μ</mi> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>6</mn> <mo>+</mo> <mn>4</mn> <mi>μ</mi> </mtd> </mtr> <mtr> <mtd> <mi>u</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>μ</mi> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>o</mtext> </mstyle> </mrow> <mspace width="2em" /> <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> <mtr> <mtd> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>u</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>7</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\begin{aligned}x&=7+3\lambda +2\mu \\y&=-\lambda \\z&=8-5\mu \\t&=-6+4\mu \\u&=-\mu \\\end{aligned}}\end{cases}}\,,\qquad {\mbox{o}}\qquad {\begin{pmatrix}x\\y\\z\\t\\u\\\end{pmatrix}}={\begin{pmatrix}7\\0\\8\\-6\\0\\\end{pmatrix}}+\lambda {\begin{pmatrix}3\\-1\\0\\0\\0\\\end{pmatrix}}+\mu {\begin{pmatrix}2\\0\\-5\\4\\-1\\\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4deccb31d0b42c07562360f0c2264eeb85f3763f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:72.847ex; height:15.843ex;" alt="{\displaystyle {\begin{cases}{\begin{aligned}x&=7+3\lambda +2\mu \\y&=-\lambda \\z&=8-5\mu \\t&=-6+4\mu \\u&=-\mu \\\end{aligned}}\end{cases}}\,,\qquad {\mbox{o}}\qquad {\begin{pmatrix}x\\y\\z\\t\\u\\\end{pmatrix}}={\begin{pmatrix}7\\0\\8\\-6\\0\\\end{pmatrix}}+\lambda {\begin{pmatrix}3\\-1\\0\\0\\0\\\end{pmatrix}}+\mu {\begin{pmatrix}2\\0\\-5\\4\\-1\\\end{pmatrix}}}"></span> </p> </td></tr> </tbody></table> <div class="mw-heading mw-heading3"><h3 id="Resolució_per_la_regla_de_Cramer"><span id="Resoluci.C3.B3_per_la_regla_de_Cramer"></span>Resolució per la regla de Cramer</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sistema_d%27equacions_lineals&action=edit&section=11" title="Modifica la secció: Resolució per la regla de Cramer"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r30997230"><div role="note" class="hatnote navigation-not-searchable">Article principal: <a href="/wiki/Regla_de_Cramer" title="Regla de Cramer">Regla de Cramer</a></div> <p>La <a href="/wiki/Regla_de_Cramer" title="Regla de Cramer">regla de Cramer</a> és un mètode de resolució per als sistemes d'equacions lineals que es basa en la utilització de <a href="/wiki/Determinant_(matem%C3%A0tiques)" title="Determinant (matemàtiques)">determinants</a>. Per exemple, la solució d'aquest sistema: </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{alignedat}{7}x&&\;+\;&&3y&&\;-\;&&2z&&\;=\;&&5&\\3x&&\;+\;&&5y&&\;+\;&&6z&&\;=\;&&7&\\2x&&\;+\;&&4y&&\;+\;&&3z&&\;=\;&&8&\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>3</mn> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>−</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>2</mn> <mi>z</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>5</mn> </mtd> <mtd /> </mtr> <mtr> <mtd> <mn>3</mn> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>5</mn> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>6</mn> <mi>z</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>7</mn> </mtd> <mtd /> </mtr> <mtr> <mtd> <mn>2</mn> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>4</mn> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>3</mn> <mi>z</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>8</mn> </mtd> <mtd /> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{7}x&&\;+\;&&3y&&\;-\;&&2z&&\;=\;&&5&\\3x&&\;+\;&&5y&&\;+\;&&6z&&\;=\;&&7&\\2x&&\;+\;&&4y&&\;+\;&&3z&&\;=\;&&8&\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ec576274d4b6ff0127ce52790ad9f71c4c2e2bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:18.27ex; height:8.843ex;" alt="{\displaystyle {\begin{alignedat}{7}x&&\;+\;&&3y&&\;-\;&&2z&&\;=\;&&5&\\3x&&\;+\;&&5y&&\;+\;&&6z&&\;=\;&&7&\\2x&&\;+\;&&4y&&\;+\;&&3z&&\;=\;&&8&\end{alignedat}}}"></span></dd></dl> <p>vindrà donada per: </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={\frac {\,\left|{\begin{matrix}{\color {red}5}&3&-2\\{\color {red}7}&5&6\\{\color {red}8}&4&3\end{matrix}}\right|\,}{\,\left|{\begin{matrix}1&3&-2\\3&5&6\\2&4&3\end{matrix}}\right|\,}},\;\;\;\;y={\frac {\,\left|{\begin{matrix}1&{\color {red}5}&-2\\3&{\color {red}7}&6\\2&{\color {red}8}&3\end{matrix}}\right|\,}{\,\left|{\begin{matrix}1&3&-2\\3&5&6\\2&4&3\end{matrix}}\right|\,}},\;\;\;\;z={\frac {\,\left|{\begin{matrix}1&3&{\color {red}5}\\3&5&{\color {red}7}\\2&4&{\color {red}8}\end{matrix}}\right|\,}{\,\left|{\begin{matrix}1&3&-2\\3&5&6\\2&4&3\end{matrix}}\right|\,}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>5</mn> </mstyle> </mrow> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>7</mn> </mstyle> </mrow> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>8</mn> </mstyle> </mrow> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> </mrow> <mrow> <mspace width="thinmathspace" /> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>5</mn> </mstyle> </mrow> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>7</mn> </mstyle> </mrow> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>8</mn> </mstyle> </mrow> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> </mrow> <mrow> <mspace width="thinmathspace" /> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>5</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>7</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>8</mn> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> </mrow> <mrow> <mspace width="thinmathspace" /> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {\,\left|{\begin{matrix}{\color {red}5}&3&-2\\{\color {red}7}&5&6\\{\color {red}8}&4&3\end{matrix}}\right|\,}{\,\left|{\begin{matrix}1&3&-2\\3&5&6\\2&4&3\end{matrix}}\right|\,}},\;\;\;\;y={\frac {\,\left|{\begin{matrix}1&{\color {red}5}&-2\\3&{\color {red}7}&6\\2&{\color {red}8}&3\end{matrix}}\right|\,}{\,\left|{\begin{matrix}1&3&-2\\3&5&6\\2&4&3\end{matrix}}\right|\,}},\;\;\;\;z={\frac {\,\left|{\begin{matrix}1&3&{\color {red}5}\\3&5&{\color {red}7}\\2&4&{\color {red}8}\end{matrix}}\right|\,}{\,\left|{\begin{matrix}1&3&-2\\3&5&6\\2&4&3\end{matrix}}\right|\,}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83f3666ea78df5a348f834cf7996a690f0c3876c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:61.533ex; height:19.509ex;" alt="{\displaystyle x={\frac {\,\left|{\begin{matrix}{\color {red}5}&3&-2\\{\color {red}7}&5&6\\{\color {red}8}&4&3\end{matrix}}\right|\,}{\,\left|{\begin{matrix}1&3&-2\\3&5&6\\2&4&3\end{matrix}}\right|\,}},\;\;\;\;y={\frac {\,\left|{\begin{matrix}1&{\color {red}5}&-2\\3&{\color {red}7}&6\\2&{\color {red}8}&3\end{matrix}}\right|\,}{\,\left|{\begin{matrix}1&3&-2\\3&5&6\\2&4&3\end{matrix}}\right|\,}},\;\;\;\;z={\frac {\,\left|{\begin{matrix}1&3&{\color {red}5}\\3&5&{\color {red}7}\\2&4&{\color {red}8}\end{matrix}}\right|\,}{\,\left|{\begin{matrix}1&3&-2\\3&5&6\\2&4&3\end{matrix}}\right|\,}}.}"></span></dd></dl> <p>Per a cada incògnita, el denominador és el <a href="/wiki/Determinant_(matem%C3%A0tiques)" title="Determinant (matemàtiques)">determinant</a> de la <a href="/wiki/Matriu_(matem%C3%A0tiques)" title="Matriu (matemàtiques)">matriu</a> de coeficients, mentre que el numerador és el determinant d'una matriu a la que una columna ha estat substituïda pel vector de termes constants (en vermell a les expressions anteriors). Tot i que la regla de Cramer és una aportació teòrica important i és útil per a sistemes petits, és poc pràctica per a matrius grans, ja que el càlcul de grans determinants és una mica incòmode. </p> <div class="mw-heading mw-heading3"><h3 id="Algorismes_alternatius">Algorismes alternatius</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sistema_d%27equacions_lineals&action=edit&section=12" title="Modifica la secció: Algorismes alternatius"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>S'han desenvolupat algorismes alternatius molt més eficients que el mètode de reducció de Gauss per a una gran quantitat de casos específics. La majoria d'aquests algorismes millorats tenen una complexitat computacional de <a href="/wiki/Cota_superior_asimpt%C3%B2tica" title="Cota superior asimptòtica"><i>O</i>(<i>n</i>²)</a>. Alguns dels mètodes més utilitzats són els següents: </p> <ul><li>Per als problemes 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 Ax=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>=</mo> <mi>b</mi> </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/c294fb03a23c833d5b3cc6b3cbe40f25f0005745" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.169ex; height:2.176ex;" alt="{\displaystyle Ax=b}"></span>, on <span class="mwe-math-element"><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> és una <a href="/w/index.php?title=Matriu_Toeplitz&action=edit&redlink=1" class="new" title="Matriu Toeplitz (encara no existeix)">matriu Toeplitz</a> simètrica, és possible utilitzar la <a href="/w/index.php?title=Recursi%C3%B3_de_Levinson&action=edit&redlink=1" class="new" title="Recursió de Levinson (encara no existeix)">recursió de Levinson</a> o algun dels mètodes derivats d'aquest. Un mètode derivat de la recursió de Levinson és la <a href="/w/index.php?title=Recursi%C3%B3_de_Schur&action=edit&redlink=1" class="new" title="Recursió de Schur (encara no existeix)">recursió de Schur</a>, que es fa servir àmpliament en el camp del <a href="/wiki/Processament_digital_de_senyals" class="mw-redirect" title="Processament digital de senyals">processament digital de senyals</a>.</li></ul> <ul><li>Per als problemes 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 Ax=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>=</mo> <mi>b</mi> </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/c294fb03a23c833d5b3cc6b3cbe40f25f0005745" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.169ex; height:2.176ex;" alt="{\displaystyle Ax=b}"></span>, on <span class="mwe-math-element"><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> és una <a href="/wiki/Matriu_singular" class="mw-redirect" title="Matriu singular">matriu singular</a> o gairebé singular, la matriu <span class="mwe-math-element"><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> es descompon en el producte de tres matrius én un procés que s'anomena <a href="/wiki/Descomposici%C3%B3_en_valors_singulars" title="Descomposició en valors singulars">descomposició en valors singulars</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Sistemes_homogenis">Sistemes homogenis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sistema_d%27equacions_lineals&action=edit&section=13" title="Modifica la secció: Sistemes homogenis"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si els <i>termes independents</i> del sistema són tots zero, </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\begin{aligned}\alpha _{1}^{1}x^{1}+\alpha _{2}^{1}x^{2}+\cdots +\alpha _{m}^{1}x^{m}&=0\\\alpha _{1}^{2}x^{1}+\alpha _{2}^{2}x^{2}+\cdots +\alpha _{m}^{2}x^{m}&=0\\\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots &\ldots \\\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots &\ldots \\\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots &\ldots \\\alpha _{1}^{n}x^{1}+\alpha _{2}^{n}x^{2}+\cdots +\alpha _{m}^{n}x^{m}&=0\\\end{aligned}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> </mtd> <mtd> <mi></mi> <mo>…</mo> </mtd> </mtr> <mtr> <mtd> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> </mtd> <mtd> <mi></mi> <mo>…</mo> </mtd> </mtr> <mtr> <mtd> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> <mo>…</mo> </mtd> <mtd> <mi></mi> <mo>…</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\begin{aligned}\alpha _{1}^{1}x^{1}+\alpha _{2}^{1}x^{2}+\cdots +\alpha _{m}^{1}x^{m}&=0\\\alpha _{1}^{2}x^{1}+\alpha _{2}^{2}x^{2}+\cdots +\alpha _{m}^{2}x^{m}&=0\\\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots &\ldots \\\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots &\ldots \\\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots &\ldots \\\alpha _{1}^{n}x^{1}+\alpha _{2}^{n}x^{2}+\cdots +\alpha _{m}^{n}x^{m}&=0\\\end{aligned}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ce62f2f6142268dec0dac7d3f78c2f777a1c50e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:35.422ex; height:19.176ex;" alt="{\displaystyle {\begin{cases}{\begin{aligned}\alpha _{1}^{1}x^{1}+\alpha _{2}^{1}x^{2}+\cdots +\alpha _{m}^{1}x^{m}&=0\\\alpha _{1}^{2}x^{1}+\alpha _{2}^{2}x^{2}+\cdots +\alpha _{m}^{2}x^{m}&=0\\\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots &\ldots \\\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots &\ldots \\\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots &\ldots \\\alpha _{1}^{n}x^{1}+\alpha _{2}^{n}x^{2}+\cdots +\alpha _{m}^{n}x^{m}&=0\\\end{aligned}}\end{cases}}}"></span> </p> </td></tr> </tbody></table> <p>el sistema es diu <i>homogeni</i>. Naturalment, en aquest cas, el rang de la matriu del sistema i el rang de la matriu ampliada coïncideixen i, així, un sistema homogeni és sempre <i>compatible</i> i té, com a mínim, la solució trivial </p> <table style="width:100%" border="0" cellpadding="2"> <tbody><tr> <td align="center"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{1}=x^{2}=\cdots =x^{m}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>⋯</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{1}=x^{2}=\cdots =x^{m}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf65cac1681b886a6239b5f96ae68a6e418874bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:24.052ex; height:2.676ex;" alt="{\displaystyle x^{1}=x^{2}=\cdots =x^{m}=0}"></span> </p> </td></tr> </tbody></table> <p>Solucionar un sistema homogeni, en el context de les aplicacions lineals, consisteix a trobar els vectors <span class="mwe-math-element"><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> pels quals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi {x}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi {x}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/716b80e3a30530c43680917cd49a3840d171a775" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.111ex; height:2.676ex;" alt="{\displaystyle \varphi {x}=0}"></span>, és a dir, trobar el <a href="/wiki/Nucli_(matem%C3%A0tiques)" title="Nucli (matemàtiques)">nucli</a> de l'aplicació lineal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>. Si el rang de la matriu del sistema és <span class="mwe-math-element"><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>, el nombre d'incògnites, aleshores els vectors que la componen són linealment independents i són una base de la imatge de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>. Aleshores, l'aplicació <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> és <a href="/wiki/Injectiva" class="mw-redirect" title="Injectiva">injectiva</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 \ker \varphi =\left\{0\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ker</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow> <mo>{</mo> <mn>0</mn> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ker \varphi =\left\{0\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85c7e55010def9f7054b9a0bc5f5294a5fbad07b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.665ex; height:2.843ex;" alt="{\displaystyle \ker \varphi =\left\{0\right\}}"></span> o <span class="mwe-math-element"><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 {Nuc} \varphi =\left\{0\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Nuc</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow> <mo>{</mo> <mn>0</mn> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Nuc} \varphi =\left\{0\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d57ee0c5d33097bc6df10c12758c0259937194" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.561ex; height:2.843ex;" alt="{\displaystyle \operatorname {Nuc} \varphi =\left\{0\right\}}"></span> i la solució és única: el sistema és <i>determinat</i> i l'única solució és la solució trivial. Si el rang és més petit, el sistema és <i>indeterminat</i>, perquè el nucli de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> no és trivial. </p><p>La relació entre els rangs de la matriu i de la matriu ampliada i la compatibilitat i indeterminació del sistema, així com el nombre de graus de llibertat de les solucions que hem anat trobant, se sistematitzen en l'enunciat del <a href="/wiki/Teorema_de_Rouch%C3%A9-Frobenius" title="Teorema de Rouché-Frobenius">teorema de Rouché-Frobenius</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Notes_i_referències"><span id="Notes_i_refer.C3.A8ncies"></span>Notes i referències</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sistema_d%27equacions_lineals&action=edit&section=14" title="Modifica la secció: Notes i referències"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist {{#if: | references-column-count references-column-count-{{{col}}}" style="list-style-type: decimal;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Introducció-1"><span class="mw-cite-backlink"><a href="#cite_ref-Introducció_1-0">↑</a></span> <span class="reference-text">Tal com s'explica a l'article, l'<a href="/wiki/%C3%80lgebra_lineal" title="Àlgebra lineal">àlgebra lineal</a> és una disciplina matemàtica molt ben estudiada que compta amb una gran quantitat de fonts. Es pot trobar gairebé tot el material presentat en aquest article a <a href="#Bibliografia">les obres de Lay 2005, Meyer 2001 i Strang 2005</a>.</span> </li> <li id="cite_note-Cholesky-2"><span class="mw-cite-backlink"><a href="#cite_ref-Cholesky_2-0">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFPress1992"><span style="font-variant: small-caps;">Press</span>, William H.; Saul A. Teukolsky, William T. Vetterling i Brian P. Flannery. <a rel="nofollow" class="external text" href="http://www.nr.com/"><i>Numerical Recipes a C: The Art of Scientific Computing (second edition)</i></a> (en anglès).  <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, 1992, p. 994. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/0-521-43108-5" title="Especial:Fonts bibliogràfiques/0-521-43108-5">ISBN 0-521-43108-5</a></span> [Consulta: 23 abril 2011].</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Recipes+a+C%3A+The+Art+of+Scientific+Computing+%28second+edition%29&rft.aulast=Press&rft.aufirst=William+H.&rft.date=1992&rft.pub=%5B%5BCambridge+University+Press%5D%5D&rft.pages=994&rft.isbn=0-521-43108-5&rft_id=http%3A%2F%2Fwww.nr.com%2F"><span style="display: none;"> </span></span> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180825110950/https://www.nr.com/">Arxivat</a> 2018-08-25 a <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</span> </li> </ol></div></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=Sistema_d%27equacions_lineals&action=edit&section=15" title="Modifica la secció: Bibliografia"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation book" style="font-style:normal" id="CITEREFAxler1997"><span style="font-variant: small-caps;">Axler</span>, Sheldon Jay. <i>Linear Algebra Done Right</i> (en anglès). 2a edició.  Springer-Verlag, 1997. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/0387982590" title="Especial:Fonts bibliogràfiques/0387982590">ISBN 0387982590</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+Done+Right&rft.aulast=Axler&rft.aufirst=Sheldon+Jay&rft.date=1997&rft.edition=2a+edici%C3%B3&rft.pub=Springer-Verlag&rft.isbn=0387982590"><span style="display: none;"> </span></span></li> <li><span class="citation book" style="font-style:normal" id="CITEREFLay2005"><span style="font-variant: small-caps;">Lay</span>, David C. <a rel="nofollow" class="external text" href="http://books.google.cat/books?id=e7FJM6aqZD8C&pg=PA300&lpg=PA300&dq=%22Linear+Algebra+and+Its+Applications%22&source=bl&ots=XwxRMZkVgv&sig=YsuusvEfBITXjV1_UzjmScYKgkI&hl=ca&ei=VPGyTfe4K9C4hAeY85zkDw&sa=X&oi=book_result&ct=result&resnum=8&ved=0CFcQ6AEwBw#v=onepage&q&f=false"><i>Linear Algebra and Its Applications</i></a> (en anglès). 3a edició.  Addison Wesley, 22 d'agost del 2005. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/978-0321287137" title="Especial:Fonts bibliogràfiques/978-0321287137">ISBN 978-0321287137</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+Its+Applications&rft.aulast=Lay&rft.aufirst=David+C.&rft.date=22+d%27agost+del+2005&rft.edition=3a+edici%C3%B3&rft.pub=Addison+Wesley&rft.isbn=978-0321287137&rft_id=http%3A%2F%2Fbooks.google.cat%2Fbooks%3Fid%3De7FJM6aqZD8C%26pg%3DPA300%26lpg%3DPA300%26dq%3D%2522Linear%2BAlgebra%2Band%2BIts%2BApplications%2522%26source%3Dbl%26ots%3DXwxRMZkVgv%26sig%3DYsuusvEfBITXjV1_UzjmScYKgkI%26hl%3Dca%26ei%3DVPGyTfe4K9C4hAeY85zkDw%26sa%3DX%26oi%3Dbook_result%26ct%3Dresult%26resnum%3D8%26ved%3D0CFcQ6AEwBw%23v%3Donepage%26q%26f%3Dfalse"><span style="display: none;"> </span></span></li> <li><span class="citation book" style="font-style:normal" id="CITEREFMeyer2001"><span style="font-variant: small-caps;">Meyer</span>, Carl D. <a rel="nofollow" class="external text" href="http://www.matrixanalysis.com/DownloadChapters.html"><i>Matrix Analysis and Applied Linear Algebra</i></a> (en anglès).  Society for Industrial and Applied Mathematics (SIAM), 15 de febrer del 2001. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/978-0898714548" title="Especial:Fonts bibliogràfiques/978-0898714548">ISBN 978-0898714548</a></span> [Consulta: 23 abril 2011].</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Analysis+and+Applied+Linear+Algebra&rft.aulast=Meyer&rft.aufirst=Carl+D.&rft.date=15+de+febrer+del+2001&rft.pub=Society+for+Industrial+and+Applied+Mathematics+%28SIAM%29&rft.isbn=978-0898714548&rft_id=http%3A%2F%2Fwww.matrixanalysis.com%2FDownloadChapters.html"><span style="display: none;"> </span></span> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091031193126/http://matrixanalysis.com/DownloadChapters.html">Arxivat</a> 2009-10-31 a <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</li> <li><span class="citation book" style="font-style:normal" id="CITEREFPoole2006"><span style="font-variant: small-caps;">Poole</span>, David. <i>Linear Algebra: A Modern Introduction</i> (en anglès). 2a edició.  Brooks/Cole, 2006. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/0-534-99845-3" title="Especial:Fonts bibliogràfiques/0-534-99845-3">ISBN 0-534-99845-3</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra%3A+A+Modern+Introduction&rft.aulast=Poole&rft.aufirst=David&rft.date=2006&rft.edition=2a+edici%C3%B3&rft.pub=Brooks%2FCole&rft.isbn=0-534-99845-3"><span style="display: none;"> </span></span></li> <li><span class="citation book" style="font-style:normal" id="CITEREFAnton2005"><span style="font-variant: small-caps;">Anton</span>, Howard. <i>Elementary Linear Algebra (Applications Version)</i> (en anglès). 9a edició.  Wiley International, 2005.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Linear+Algebra+%28Applications+Version%29&rft.aulast=Anton&rft.aufirst=Howard&rft.date=2005&rft.edition=9a+edici%C3%B3&rft.pub=Wiley+International"><span style="display: none;"> </span></span></li> <li><span class="citation book" style="font-style:normal" id="CITEREFLeon2006"><span style="font-variant: small-caps;">Leon</span>, Steven J. <a rel="nofollow" class="external text" href="http://books.google.cat/books?id=GmpV5YH4Mv8C&printsec=frontcover&dq=%22Linear+Algebra+With+Applications%22&source=bl&ots=5Dk_UkCzfU&sig=9iZ5wvEAEVlkIsyJntNV1oVwnnc&hl=ca&ei=5PGyTaObK4W2hAe8waTkDw&sa=X&oi=book_result&ct=result&resnum=6&ved=0CEoQ6AEwBQ#v=onepage&q&f=false"><i>Linear Algebra With Applications</i></a> (en anglès). 7a edició.  Pearson Prentice Hall, 2006.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+With+Applications&rft.aulast=Leon&rft.aufirst=Steven+J.&rft.date=2006&rft.edition=7a+edici%C3%B3&rft.pub=Pearson+Prentice+Hall&rft_id=http%3A%2F%2Fbooks.google.cat%2Fbooks%3Fid%3DGmpV5YH4Mv8C%26printsec%3Dfrontcover%26dq%3D%2522Linear%2BAlgebra%2BWith%2BApplications%2522%26source%3Dbl%26ots%3D5Dk_UkCzfU%26sig%3D9iZ5wvEAEVlkIsyJntNV1oVwnnc%26hl%3Dca%26ei%3D5PGyTaObK4W2hAe8waTkDw%26sa%3DX%26oi%3Dbook_result%26ct%3Dresult%26resnum%3D6%26ved%3D0CEoQ6AEwBQ%23v%3Donepage%26q%26f%3Dfalse"><span style="display: none;"> </span></span></li> <li><span class="citation book" style="font-style:normal" id="CITEREFStrang2005"><span style="font-variant: small-caps;">Strang</span>, Gilbert. <i>Linear Algebra and Its Applications</i> (en anglès), 2005.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+Its+Applications&rft.aulast=Strang&rft.aufirst=Gilbert&rft.date=2005"><span style="display: none;"> </span></span></li></ul> <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=Sistema_d%27equacions_lineals&action=edit&section=16" title="Modifica la secció: Vegeu també"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Sistema_d%27equacions" title="Sistema d'equacions">Sistema d'equacions</a></li> <li><a href="/wiki/Zhang_Qiujian" title="Zhang Qiujian">Zhang Qiujian</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Enllaços_externs"><span id="Enlla.C3.A7os_externs"></span>Enllaços externs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sistema_d%27equacions_lineals&action=edit&section=17" title="Modifica la secció: Enllaços externs"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://quadernmat.blogspot.com/2009/05/sistemes.html">"Sistemes d'equacions lineals"</a> a <i>Quadern de matemàtiques</i>.</li> <li><a rel="nofollow" class="external text" href="http://dmi.uib.es/~mmoya/Docencia/Biologia/Preliminars/sistemes.pdf">Resolució de sistemes d'equacions lineals</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160303190811/http://dmi.uib.es/~mmoya/Docencia/Biologia/Preliminars/sistemes.pdf">Arxivat</a> 2016-03-03 a <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. (PDF)</li> <li><a rel="nofollow" class="external text" href="http://www.solvingequations.net/">Resolució de sistemes lineals en línia</a> <style data-mw-deduplicate="TemplateStyles:r33711417">.mw-parser-output .languageicon{font-size:0.95em;color:#555;background-color:inherit}@media screen{html.skin-theme-clientpref-night .mw-parser-output .languageicon{background-color:inherit;color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .languageicon{background-color:inherit;color:white}}</style><span class="languageicon" title="En anglès">(anglès)</span></li> <li><a rel="nofollow" class="external text" href="http://wims.unice.fr/wims/wims.cgi?session=Q70382836C.2&+lang=en&+module=U1/algebra/visgauss.en">Exercicis interactius, pas a pas, del mètode de reducció de Gauss per a matrius i sistemes d'equacions lineals, en cossos diversos</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33711417"><span class="languageicon" title="En anglès">(anglès)</span></li> <li><a rel="nofollow" class="external text" href="http://wims.unice.fr/wims/wims.cgi?module=tool/linear/linsolver.en">Resolució de sistemes d'equacions lineals</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33711417"><span class="languageicon" title="En anglès">(anglès)</span></li></ul> <p><br /> </p><p><br /> </p> <div role="navigation" class="navbox" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Control_d%27autoritats" title="Control d'autoritats">Registres d'autoritat</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a> <span class="uid"> (<a rel="nofollow" class="external text" href="http://d-nb.info/gnd/4035826-4">1</a>)</span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Bases d'informació</th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/GEC" class="mw-redirect" title="GEC">GEC</a> <span class="uid"> (<a rel="nofollow" class="external text" href="https://www.enciclopedia.cat/gran-enciclopedia-catalana/sistema-dequacions-lineals">1</a>)</span></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtingut de «<a dir="ltr" href="https://ca.wikipedia.org/w/index.php?title=Sistema_d%27equacions_lineals&oldid=32110933">https://ca.wikipedia.org/w/index.php?title=Sistema_d%27equacions_lineals&oldid=32110933</a>»</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categories</a>: <ul><li><a href="/wiki/Categoria:Equacions" title="Categoria:Equacions">Equacions</a></li><li><a href="/wiki/Categoria:Matrius" title="Categoria:Matrius">Matrius</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categories ocultes: <ul><li><a href="/wiki/Categoria:Articles_amb_la_plantilla_Webarchive_amb_enlla%C3%A7_wayback" title="Categoria:Articles amb la plantilla Webarchive amb enllaç wayback">Articles amb la plantilla Webarchive amb enllaç wayback</a></li><li><a href="/wiki/Categoria:1.000_articles_fonamentals" title="Categoria:1.000 articles fonamentals">1.000 articles fonamentals</a></li><li><a href="/wiki/Categoria:Ci%C3%A8ncia_(Els_1000_de_META)" title="Categoria:Ciència (Els 1000 de META)">Ciència (Els 1000 de META)</a></li><li><a href="/wiki/Categoria:Control_d%27autoritats" title="Categoria:Control d'autoritats">Control d'autoritats</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> La pàgina va ser modificada per darrera vegada el 30 juny 2023 a les 00:07.</li> <li id="footer-info-copyright">El text està disponible sota la <a href="/wiki/Viquip%C3%A8dia:Text_de_la_llic%C3%A8ncia_de_Creative_Commons_Reconeixement-Compartir_Igual_4.0_No_adaptada" title="Viquipèdia:Text de la llicència de Creative Commons Reconeixement-Compartir Igual 4.0 No adaptada"> Llicència de Creative Commons Reconeixement i Compartir-Igual</a>; es poden aplicar termes addicionals. 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