线性方程组 - 维基百科，自由的百科全书

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class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>创建账号</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84" title="建议你登录，尽管并非必须。[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>登录</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</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">序言</div> </a> </li> <li id="toc-例子" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#例子"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>例子</span> </div> </a> <ul id="toc-例子-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-线性方程组的解" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#线性方程组的解"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>线性方程组的解</span> </div> </a> <button aria-controls="toc-线性方程组的解-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>开关线性方程组的解子章节</span> </button> <ul id="toc-线性方程组的解-sublist" class="vector-toc-list"> <li id="toc-几何解释" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#几何解释"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>几何解释</span> </div> </a> <ul id="toc-几何解释-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-齐次线性方程组" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#齐次线性方程组"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>齐次线性方程组</span> </div> </a> <ul id="toc-齐次线性方程组-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-松弛求解" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#松弛求解"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>松弛求解</span> </div> </a> <ul id="toc-松弛求解-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-求解" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#求解"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>求解</span> </div> </a> <button aria-controls="toc-求解-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>开关求解子章节</span> </button> <ul id="toc-求解-sublist" class="vector-toc-list"> <li id="toc-克莱姆法则" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#克莱姆法则"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>克莱姆法则</span> </div> </a> <ul id="toc-克莱姆法则-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-数值方法" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#数值方法"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>数值方法</span> </div> </a> <ul id="toc-数值方法-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-应用" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#应用"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>应用</span> </div> </a> <ul id="toc-应用-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-相关软件" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#相关软件"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>相关软件</span> </div> </a> <ul id="toc-相关软件-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-其他方法与软件" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#其他方法与软件"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>其他方法与软件</span> </div> </a> <ul id="toc-其他方法与软件-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参见" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参见"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>参见</span> </div> </a> <ul id="toc-参见-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参考文献" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参考文献"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>参考文献</span> </div> </a> <ul id="toc-参考文献-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部連結" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部連結"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>外部連結</span> </div> </a> <ul id="toc-外部連結-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="目录" 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="开关目录" > <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">开关目录</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">线性方程组</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="前往另一种语言写成的文章。70种语言可用" > <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种语言</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 – 瑞士德语" lang="gsw" hreflang="gsw" data-title="Lineares Gleichungssystem" data-language-autonym="Alemannisch" data-language-local-name="瑞士德语" 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 – 阿拉贡语" lang="an" hreflang="an" data-title="Sistema d'ecuacions lineals" data-language-autonym="Aragonés" data-language-local-name="阿拉贡语" 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 – 阿拉伯语" lang="ar" hreflang="ar" data-title="نظام معادلات خطية" data-language-autonym="العربية" data-language-local-name="阿拉伯语" 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 – 阿斯图里亚斯语" lang="ast" hreflang="ast" data-title="Sistema d'ecuaciones lliniales" data-language-autonym="Asturianu" data-language-local-name="阿斯图里亚斯语" 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 – 阿塞拜疆语" lang="az" hreflang="az" data-title="Xətti tənliklər sistemi" data-language-autonym="Azərbaycanca" data-language-local-name="阿塞拜疆语" 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 – 巴什基尔语" lang="ba" hreflang="ba" data-title="Һыҙыҡлы алгебраик тигеҙләмәләр системаһы" data-language-autonym="Башҡортса" data-language-local-name="巴什基尔语" 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 – 白俄罗斯语" lang="be" hreflang="be" data-title="Сістэма лінейных алгебраічных ураўненняў" data-language-autonym="Беларуская" data-language-local-name="白俄罗斯语" 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 – 保加利亚语" lang="bg" hreflang="bg" data-title="Система линейни уравнения" data-language-autonym="Български" data-language-local-name="保加利亚语" 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 – 波斯尼亚语" lang="bs" hreflang="bs" data-title="Sistem linearnih jednačina" data-language-autonym="Bosanski" data-language-local-name="波斯尼亚语" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Sistema_d%27equacions_lineals" title="Sistema d'equacions lineals – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Sistema d'equacions lineals" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%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 – 中库尔德语" lang="ckb" hreflang="ckb" data-title="سیستمی ھاوکێشەی ھێڵی" data-language-autonym="کوردی" data-language-local-name="中库尔德语" 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 – 捷克语" lang="cs" hreflang="cs" data-title="Soustava lineárních rovnic" data-language-autonym="Čeština" data-language-local-name="捷克语" 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 – 楚瓦什语" lang="cv" hreflang="cv" data-title="Линилле алгебрăлла танлăхсен тытăмĕ" data-language-autonym="Чӑвашла" data-language-local-name="楚瓦什语" 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 – 德语" lang="de" hreflang="de" data-title="Lineares Gleichungssystem" data-language-autonym="Deutsch" data-language-local-name="德语" 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 – 希腊语" lang="el" hreflang="el" data-title="Σύστημα γραμμικών εξισώσεων" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" 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 – 英语" lang="en" hreflang="en" data-title="System of linear equations" data-language-autonym="English" data-language-local-name="英语" 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 – 世界语" lang="eo" hreflang="eo" data-title="Sistemo de linearaj ekvacioj" data-language-autonym="Esperanto" data-language-local-name="世界语" 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 – 西班牙语" lang="es" hreflang="es" data-title="Sistema de ecuaciones lineales" data-language-autonym="Español" data-language-local-name="西班牙语" 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 – 爱沙尼亚语" lang="et" hreflang="et" data-title="Lineaarvõrrandisüsteem" data-language-autonym="Eesti" data-language-local-name="爱沙尼亚语" 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 – 巴斯克语" lang="eu" hreflang="eu" data-title="Ekuazio linealetako sistema" data-language-autonym="Euskara" data-language-local-name="巴斯克语" 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 – 波斯语" lang="fa" hreflang="fa" data-title="دستگاه معادلات خطی" data-language-autonym="فارسی" data-language-local-name="波斯语" 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 – 芬兰语" lang="fi" hreflang="fi" data-title="Lineaarinen yhtälöryhmä" data-language-autonym="Suomi" data-language-local-name="芬兰语" 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 – 法语" lang="fr" hreflang="fr" data-title="Système d'équations linéaires" data-language-autonym="Français" data-language-local-name="法语" 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 – 加利西亚语" lang="gl" hreflang="gl" data-title="Sistema de ecuacións lineais" data-language-autonym="Galego" data-language-local-name="加利西亚语" 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 – 希伯来语" lang="he" hreflang="he" data-title="מערכת משוואות ליניאריות" data-language-autonym="עברית" data-language-local-name="希伯来语" 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 – 印地语" lang="hi" hreflang="hi" data-title="रैखिक समीकरण निकाय" data-language-autonym="हिन्दी" data-language-local-name="印地语" 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 – 斐濟印地文" lang="hif" hreflang="hif" data-title="System of linear equations" data-language-autonym="Fiji Hindi" data-language-local-name="斐濟印地文" 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 – 克罗地亚语" lang="hr" hreflang="hr" data-title="Sustav linearnih jednadžbi" data-language-autonym="Hrvatski" data-language-local-name="克罗地亚语" 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 – 匈牙利语" lang="hu" hreflang="hu" data-title="Lineáris egyenletrendszer" data-language-autonym="Magyar" data-language-local-name="匈牙利语" 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 – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Գծային հավասարումների համակարգ" data-language-autonym="Հայերեն" data-language-local-name="亚美尼亚语" 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 – 国际语" lang="ia" hreflang="ia" data-title="Systema de equationes linear" data-language-autonym="Interlingua" data-language-local-name="国际语" 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 – 印度尼西亚语" lang="id" hreflang="id" data-title="Sistem persamaan linear" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" 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 – 冰岛语" lang="is" hreflang="is" data-title="Línulegt jöfnuhneppi" data-language-autonym="Íslenska" data-language-local-name="冰岛语" 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 – 意大利语" lang="it" hreflang="it" data-title="Sistema di equazioni lineari" data-language-autonym="Italiano" data-language-local-name="意大利语" 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 – 日语" lang="ja" hreflang="ja" data-title="線型方程式系" data-language-autonym="日本語" data-language-local-name="日语" 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 – 韩语" lang="ko" hreflang="ko" data-title="연립 일차 방정식" data-language-autonym="한국어" data-language-local-name="韩语" 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 – 拉丁语" lang="la" hreflang="la" data-title="Systema aequationum linearium" data-language-autonym="Latina" data-language-local-name="拉丁语" 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 – 倫巴底文" lang="lmo" hreflang="lmo" data-title="Sistema de equazion linear" data-language-autonym="Lombard" data-language-local-name="倫巴底文" 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 – 拉脱维亚语" lang="lv" hreflang="lv" data-title="Lineāru vienādojumu sistēma" data-language-autonym="Latviešu" data-language-local-name="拉脱维亚语" 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 – 马其顿语" lang="mk" hreflang="mk" data-title="Систем на линеарни равенки" data-language-autonym="Македонски" data-language-local-name="马其顿语" 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 – 马来语" lang="ms" hreflang="ms" data-title="Sistem persamaan linear" data-language-autonym="Bahasa Melayu" data-language-local-name="马来语" 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 – 荷兰语" lang="nl" hreflang="nl" data-title="Stelsel van lineaire vergelijkingen" data-language-autonym="Nederlands" data-language-local-name="荷兰语" 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 – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Lineært likningssystem" data-language-autonym="Norsk nynorsk" data-language-local-name="挪威尼诺斯克语" 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 – 书面挪威语" lang="nb" hreflang="nb" data-title="Lineært ligningssystem" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" 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 – 奥克语" lang="oc" hreflang="oc" data-title="Sistèma d'equacions linearas" data-language-autonym="Occitan" data-language-local-name="奥克语" 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 – 旁遮普语" lang="pa" hreflang="pa" data-title="ਰੇਖੀ ਸਮੀਕਰਨਾਂ ਦਾ ਤੰਤਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="旁遮普语" 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 – 波兰语" lang="pl" hreflang="pl" data-title="Układ równań liniowych" data-language-autonym="Polski" data-language-local-name="波兰语" 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 – 葡萄牙语" lang="pt" hreflang="pt" data-title="Sistema de equações lineares" data-language-autonym="Português" data-language-local-name="葡萄牙语" 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 – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Sistem de ecuații liniare" data-language-autonym="Română" data-language-local-name="罗马尼亚语" 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 – 俄语" lang="ru" hreflang="ru" data-title="Система линейных алгебраических уравнений" data-language-autonym="Русский" data-language-local-name="俄语" 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 – 信德语" lang="sd" hreflang="sd" data-title="سِڌِر مساواتُن جو سرشتو" data-language-autonym="سنڌي" data-language-local-name="信德语" 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 – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Sistem linearnih jednačina" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞尔维亚-克罗地亚语" 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 – 僧伽罗语" lang="si" hreflang="si" data-title="ඒකජ සමීකරණ පද්ධතිය" data-language-autonym="සිංහල" data-language-local-name="僧伽罗语" 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 – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Sústava lineárnych rovníc" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克语" 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 – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Sistem linearnih enačb" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" 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 – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Систем линеарних једначина" data-language-autonym="Српски / srpski" data-language-local-name="塞尔维亚语" 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 – 瑞典语" lang="sv" hreflang="sv" data-title="Linjärt ekvationssystem" data-language-autonym="Svenska" data-language-local-name="瑞典语" 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 – 泰米尔语" lang="ta" hreflang="ta" data-title="நேரியல் சமன்பாடுகளின் தொகுப்பு" data-language-autonym="தமிழ்" data-language-local-name="泰米尔语" 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 – 泰语" lang="th" hreflang="th" data-title="ระบบสมการเชิงเส้น" data-language-autonym="ไทย" data-language-local-name="泰语" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Do%C4%9Frusal_denklem_dizgesi" title="Doğrusal denklem dizgesi – 土耳其语" lang="tr" hreflang="tr" data-title="Doğrusal denklem dizgesi" data-language-autonym="Türkçe" data-language-local-name="土耳其语" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A1%D1%8B%D0%B7%D1%8B%D0%BA%D1%87%D0%B0_%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%D0%B7%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%D1%81%D1%8B" title="Сызыкча алгебраик тигезләмәләр системасы – 鞑靼语" lang="tt" hreflang="tt" data-title="Сызыкча алгебраик тигезләмәләр системасы" data-language-autonym="Татарча / tatarça" data-language-local-name="鞑靼语" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BB%D1%96%D0%BD%D1%96%D0%B9%D0%BD%D0%B8%D1%85_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D1%97%D1%87%D0%BD%D0%B8%D1%85_%D1%80%D1%96%D0%B2%D0%BD%D1%8F%D0%BD%D1%8C" title="Система лінійних алгебраїчних рівнянь – 乌克兰语" lang="uk" hreflang="uk" data-title="Система лінійних алгебраїчних рівнянь" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DB%8C%DA%A9%D9%84%D8%AE%D8%AA_%D9%84%DA%A9%DB%8C%D8%B1%DB%8C_%D9%85%D8%B3%D8%A7%D9%88%D8%A7%D8%AA_%DA%A9%D8%A7_%D9%86%D8%B8%D8%A7%D9%85" title="یکلخت لکیری مساوات کا نظام – 乌尔都语" lang="ur" hreflang="ur" data-title="یکلخت لکیری مساوات کا نظام" data-language-autonym="اردو" data-language-local-name="乌尔都语" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%E1%BB%87_ph%C6%B0%C6%A1ng_tr%C3%ACnh_tuy%E1%BA%BFn_t%C3%ADnh" title="Hệ phương trình tuyến tính – 越南语" lang="vi" hreflang="vi" data-title="Hệ phương trình tuyến tính" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Sistema_han_ekawsyones_linyales" title="Sistema han ekawsyones linyales – 瓦瑞语" lang="war" hreflang="war" data-title="Sistema han ekawsyones linyales" data-language-autonym="Winaray" data-language-local-name="瓦瑞语" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84" title="线性方程组 – 吴语" lang="wuu" hreflang="wuu" data-title="线性方程组" data-language-autonym="吴语" data-language-local-name="吴语" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%B7%9A%E6%80%A7%E6%96%B9%E7%A8%8B%E7%B5%84" title="線性方程組 – 粤语" lang="yue" hreflang="yue" data-title="線性方程組" data-language-autonym="粵語" data-language-local-name="粤语" 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data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">外观</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-noteTA-307af9e9" class="mw-indicator"><div class="mw-parser-output"><span class="skin-invert" typeof="mw:File"><span title="本页使用了标题或全文手工转换"><img alt="本页使用了标题或全文手工转换" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/35px-Zh_conversion_icon_m.svg.png" decoding="async" width="35" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/53px-Zh_conversion_icon_m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/70px-Zh_conversion_icon_m.svg.png 2x" data-file-width="32" data-file-height="20" /></span></span></div></div> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</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="zh" dir="ltr"><div id="noteTA-307af9e9" class="noteTA"><div class="noteTA-group"><div data-noteta-group-source="module" data-noteta-group="Math"></div></div></div> <table class="infobox noprint" style="width:210px; float: right; clear: right; text-align:center; margin-top:1em;"> <tbody><tr> <th style="font-size:90%; background:#DCF0FF"><a href="/wiki/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="线性代数">线性代数</a> </th></tr> <tr> <td><div style="padding-top: 7px; padding-bottom: 4px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{bmatrix}1&2\\3&4\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{bmatrix}1&2\\3&4\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a31efc33ac33577d719a3ccd162a9bf21e4847ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.972ex; height:6.176ex;" alt="{\displaystyle \mathbf {A} ={\begin{bmatrix}1&2\\3&4\end{bmatrix}}}"></span></div> </td></tr> <tr style="font-size: 90%; line-height: 150%;"> <td><span class="nowrap"><a href="/wiki/%E5%90%91%E9%87%8F" title="向量">向量</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间">向量空间</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%9F%BA_(%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8)" title="基 (線性代數)">基底</a> <span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%A1%8C%E5%88%97%E5%BC%8F" title="行列式">行列式</a> <span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a></span> </td></tr> <tr> <td> <table class="collapsible collapsed" width="100%"> <tbody><tr> <th style="text-align: left; background: #DCF0FF; font-size: 90%;">向量 </th></tr> <tr style="font-size: 90%; line-height: 150%;"> <td><span class="nowrap"><a href="/wiki/%E6%A0%87%E9%87%8F_(%E6%95%B0%E5%AD%A6)" title="标量 (数学)">标量</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%90%91%E9%87%8F" title="向量">向量</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间">向量空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%82%B9%E7%A7%AF" title="点积">向量投影</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%A4%96%E7%A7%AF" class="mw-disambig" title="外积">外积</a>（<a href="/wiki/%E5%8F%89%E7%A7%AF" title="叉积">向量积</a> ·</span> <span class="nowrap"><a href="/wiki/%E4%B8%83%E7%BB%B4%E5%8F%89%E7%A7%AF" title="七维叉积">七维向量积</a>） ·</span> <span class="nowrap"><a href="/wiki/%E5%86%85%E7%A7%AF%E7%A9%BA%E9%97%B4" title="内积空间">内积</a>（<a href="/wiki/%E7%82%B9%E7%A7%AF" title="点积">数量积</a>） ·</span> <span class="nowrap"><a href="/wiki/%E4%BA%8C%E9%87%8D%E5%90%91%E9%87%8F" title="二重向量">二重向量</a></span> </td></tr></tbody></table> <table class="collapsible collapsed" width="100%"> <tbody><tr> <th style="text-align: left; background: #DCF0FF; font-size: 90%;">矩阵与行列式 </th></tr> <tr style="font-size: 90%; line-height: 150%;"> <td><span class="nowrap"><a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E8%A1%8C%E5%88%97%E5%BC%8F" title="行列式">行列式</a> ·</span> <span class="nowrap"><a class="mw-selflink selflink">线性方程组</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%A7%A9_(%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0)" title="秩 (线性代数)">秩</a> ·</span> <span class="nowrap"><a href="/wiki/%E9%9B%B6%E7%A9%BA%E9%97%B4" title="零空间">核</a> ·</span> <span class="nowrap"><a href="/wiki/%E8%B7%A1" title="跡">跡</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%96%AE%E4%BD%8D%E7%9F%A9%E9%99%A3" title="單位矩陣">單位矩陣</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%88%9D%E7%AD%89%E7%9F%A9%E9%98%B5" title="初等矩阵">初等矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%96%B9%E5%9D%97%E7%9F%A9%E9%98%B5" title="方块矩阵">方块矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%88%86%E5%A1%8A%E7%9F%A9%E9%99%A3" title="分塊矩陣">分块矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E4%B8%89%E8%A7%92%E7%9F%A9%E9%98%B5" title="三角矩阵">三角矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E9%9D%9E%E5%A5%87%E5%BC%82%E6%96%B9%E9%98%B5" title="非奇异方阵">非奇异方阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E8%BD%AC%E7%BD%AE%E7%9F%A9%E9%98%B5" title="转置矩阵">转置矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E9%80%86%E7%9F%A9%E9%98%B5" title="逆矩阵">逆矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%B0%8D%E8%A7%92%E7%9F%A9%E9%99%A3" title="對角矩陣">对角矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%8F%AF%E5%AF%B9%E8%A7%92%E5%8C%96%E7%9F%A9%E9%98%B5" title="可对角化矩阵">可对角化矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3" title="對稱矩陣">对称矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%8F%8D%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3" title="反對稱矩陣">反對稱矩陣</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%AD%A3%E4%BA%A4%E7%9F%A9%E9%98%B5" title="正交矩阵">正交矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E9%85%89%E7%9F%A9%E9%98%B5" title="酉矩阵">幺正矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%9F%83%E5%B0%94%E7%B1%B3%E7%89%B9%E7%9F%A9%E9%98%B5" title="埃尔米特矩阵">埃尔米特矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%96%9C%E5%9F%83%E5%B0%94%E7%B1%B3%E7%89%B9%E7%9F%A9%E9%98%B5" title="斜埃尔米特矩阵">反埃尔米特矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%AD%A3%E8%A7%84%E7%9F%A9%E9%98%B5" title="正规矩阵">正规矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E4%BC%B4%E9%9A%8F%E7%9F%A9%E9%98%B5" title="伴随矩阵">伴随矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E9%A4%98%E5%9B%A0%E5%AD%90%E7%9F%A9%E9%99%A3" title="餘因子矩陣">余因子矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%85%B1%E8%BD%AD%E8%BD%AC%E7%BD%AE" title="共轭转置">共轭转置</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%AD%A3%E5%AE%9A%E7%9F%A9%E9%98%B5" title="正定矩阵">正定矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%B9%82%E9%9B%B6%E7%9F%A9%E9%98%B5" title="幂零矩阵">幂零矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%A3" title="矩阵分解">矩阵分解</a> （<a href="/wiki/LU%E5%88%86%E8%A7%A3" title="LU分解">LU分解</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3" title="奇异值分解">奇异值分解</a> ·</span> <span class="nowrap"><a href="/wiki/QR%E5%88%86%E8%A7%A3" title="QR分解">QR分解</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%9E%81%E5%88%86%E8%A7%A3" title="极分解">极分解</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%89%B9%E5%BE%81%E5%88%86%E8%A7%A3" title="特征分解">特征分解</a>） ·</span> <span class="nowrap"><a href="/wiki/%E5%AD%90%E5%BC%8F%E5%92%8C%E4%BD%99%E5%AD%90%E5%BC%8F" title="子式和余子式">子式和余子式</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E5%B1%95%E5%BC%80" title="拉普拉斯展开">拉普拉斯展開</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%85%8B%E7%BD%97%E5%86%85%E5%85%8B%E7%A7%AF" title="克罗内克积">克罗内克积</a></span> </td></tr></tbody></table> <table class="collapsible collapsed" width="100%"> <tbody><tr> <th style="text-align: left; background: #DCF0FF; font-size: 90%;">线性空间与线性变换 </th></tr> <tr style="font-size: 90%; line-height: 150%;"> <td><span class="nowrap"><a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间">线性空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="线性映射">线性变换</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E5%AD%90%E7%A9%BA%E9%97%B4" title="线性子空间">线性子空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E7%94%9F%E6%88%90%E7%A9%BA%E9%97%B4" title="线性生成空间">线性生成空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%9F%BA_(%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8)" title="基 (線性代數)">基</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="线性映射">线性映射</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%8A%95%E5%BD%B1" class="mw-disambig" title="投影">线性投影</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%B7%9A%E6%80%A7%E7%84%A1%E9%97%9C" title="線性無關">線性無關</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E7%BB%84%E5%90%88" title="线性组合">线性组合</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%B3%9B%E5%87%BD" class="mw-redirect" title="线性泛函">线性泛函</a> ·</span> <span class="nowrap"><a href="/wiki/%E8%A1%8C%E7%A9%BA%E9%97%B4%E4%B8%8E%E5%88%97%E7%A9%BA%E9%97%B4" title="行空间与列空间">行空间与列空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%AF%B9%E5%81%B6%E7%A9%BA%E9%97%B4" title="对偶空间">对偶空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%AD%A3%E4%BA%A4" title="正交">正交</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%89%B9%E5%BE%81%E5%80%BC%E5%92%8C%E7%89%B9%E5%BE%81%E5%90%91%E9%87%8F" title="特征值和特征向量">特征向量</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98%E6%B3%95" title="最小二乘法">最小二乘法</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%A0%BC%E6%8B%89%E5%A7%86-%E6%96%BD%E5%AF%86%E7%89%B9%E6%AD%A3%E4%BA%A4%E5%8C%96" title="格拉姆-施密特正交化">格拉姆-施密特正交化</a></span> </td></tr></tbody></table> </td></tr> <tr style="text-align: center; font-size: 90%;"> <td><style data-mw-deduplicate="TemplateStyles:r84265675">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:" :"}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist-pipe dd::after,.mw-parser-output .hlist-pipe li::after{content:" | ";font-weight:normal}.mw-parser-output .hlist-hyphen dd::after,.mw-parser-output .hlist-hyphen li::after{content:" - ";font-weight:normal}.mw-parser-output .hlist-comma dd::after,.mw-parser-output .hlist-comma li::after{content:"、";font-weight:normal}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:"（"counter(listitem)"\a0 "}.mw-parser-output ul.cslist,.mw-parser-output ul.sslist{margin:0;padding:0;display:inline-block;list-style:none}.mw-parser-output .cslist li,.mw-parser-output .sslist li{margin:0;display:inline-block}.mw-parser-output .cslist li::after{content:"，"}.mw-parser-output .sslist li::after{content:"；"}.mw-parser-output .cslist li:last-child::after,.mw-parser-output .sslist li:last-child::after{content:none}</style><style data-mw-deduplicate="TemplateStyles:r84244141">.mw-parser-output .navbar{display:inline;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:110%;margin:0 8em}.mw-parser-output .navbar-ct-mini{font-size:110%;margin:0 5em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="Template:线性代数"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0&action=edit&redlink=1" class="new" title="Template talk:线性代数（页面不存在）"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="Special:编辑页面/Template:线性代数"><abbr title="编辑该模板">编</abbr></a></li></ul></div> </td></tr></tbody></table> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Secretsharing-3-point.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Secretsharing-3-point.png/220px-Secretsharing-3-point.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Secretsharing-3-point.png/330px-Secretsharing-3-point.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Secretsharing-3-point.png/440px-Secretsharing-3-point.png 2x" data-file-width="480" data-file-height="480" /></a><figcaption>三變量的線性系統確定了一組<a href="/wiki/%E5%B9%B3%E9%9D%A2_(%E6%95%B0%E5%AD%A6)" title="平面 (数学)">平面</a>。交點就是解。</figcaption></figure> <p><b>线性方程组</b>是数学<a href="/wiki/%E6%96%B9%E7%A8%8B%E7%BB%84" title="方程组">方程组</a>的一种，它符合以下的形式： </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{cases}a_{1,1}x_{1}+a_{1,2}x_{2}+\cdots +a_{1,n}x_{n}=b_{1}\\a_{2,1}x_{1}+a_{2,2}x_{2}+\cdots +a_{2,n}x_{n}=b_{2}\\\vdots \quad \quad \quad \vdots \\a_{m,1}x_{1}+a_{m,2}x_{2}+\cdots +a_{m,n}x_{n}=b_{m}\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> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <msub> <mi>x</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"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>x</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> <mo>,</mo> <mn>2</mn> </mrow> </msub> <msub> <mi>x</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"> <mn>2</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <msub> <mi>x</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> <mo>,</mo> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}a_{1,1}x_{1}+a_{1,2}x_{2}+\cdots +a_{1,n}x_{n}=b_{1}\\a_{2,1}x_{1}+a_{2,2}x_{2}+\cdots +a_{2,n}x_{n}=b_{2}\\\vdots \quad \quad \quad \vdots \\a_{m,1}x_{1}+a_{m,2}x_{2}+\cdots +a_{m,n}x_{n}=b_{m}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/225bfdbd757b1d472caf497a455c4f5085a5fb97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:39.866ex; height:12.843ex;" alt="{\displaystyle {\begin{cases}a_{1,1}x_{1}+a_{1,2}x_{2}+\cdots +a_{1,n}x_{n}=b_{1}\\a_{2,1}x_{1}+a_{2,2}x_{2}+\cdots +a_{2,n}x_{n}=b_{2}\\\vdots \quad \quad \quad \vdots \\a_{m,1}x_{1}+a_{m,2}x_{2}+\cdots +a_{m,n}x_{n}=b_{m}\end{cases}}}"></span></dd></dl> <p>其中的<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1,1},\,a_{1,2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1,1},\,a_{1,2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cfec5f6803a11a54a6a5032fac7c935fbc1093f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.548ex; height:2.343ex;" alt="{\displaystyle a_{1,1},\,a_{1,2}}"></span>以及<span class="mwe-math-element"><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_{1},\,b_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{1},\,b_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46e87c2d8504a826af2123f920ab2182759739a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.525ex; height:2.509ex;" alt="{\displaystyle b_{1},\,b_{2}}"></span>等等是已知的常数，而<span class="mwe-math-element"><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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\,x_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/844c937909907ba0a8d46d070772fab9992d1646" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.189ex; height:2.009ex;" alt="{\displaystyle x_{1},\,x_{2}}"></span>等等则是要求的未知数。 </p><p>如果用<a href="/wiki/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="线性代数">线性代数</a>中的概念来表达，则线性方程组可以写成： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} \mathbf {x} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \mathbf {x} =\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1688bdcfe95659c92f6452378ee805da3f796c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.014ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} \mathbf {x} =\mathbf {b} }"></span></dd></dl> <p>這裡的<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span>是<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span><a href="/wiki/%E7%9F%A9%E9%99%A3" class="mw-redirect" title="矩陣">矩陣</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 \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"></span>是含有<span class="mwe-math-element"><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>个元素<a href="/wiki/%E5%90%91%E9%87%8F" title="向量">列向量</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 \mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }"></span>是含有<span class="mwe-math-element"><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>个元素列向量。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\a_{2,1}&a_{2,2}&\cdots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{m,1}&a_{m,2}&\cdots &a_{m,n}\end{bmatrix}},\quad \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\quad \mathbf {b} ={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\a_{2,1}&a_{2,2}&\cdots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{m,1}&a_{m,2}&\cdots &a_{m,n}\end{bmatrix}},\quad \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\quad \mathbf {b} ={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57d432469d5c2d250696d072006022ec28b64b47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:60.365ex; height:14.843ex;" alt="{\displaystyle \mathbf {A} ={\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\a_{2,1}&a_{2,2}&\cdots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{m,1}&a_{m,2}&\cdots &a_{m,n}\end{bmatrix}},\quad \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\quad \mathbf {b} ={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}}"></span></dd></dl> <p>这是线性方程组的另一种记录方法。在已知矩阵<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span>和向量<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }"></span>的情况求得未知向量<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"></span>是<a href="/wiki/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="线性代数">线性代数</a>的基本问题之一。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="例子"><span id=".E4.BE.8B.E5.AD.90"></span>例子</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84&action=edit&section=1" title="编辑章节：例子"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>以下是一个由两个方程构成的线性方程组： </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{cases}3x_{1}+5x_{2}=4\\x_{1}+2x_{2}=1\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> <mn>3</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>5</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}3x_{1}+5x_{2}=4\\x_{1}+2x_{2}=1\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a851a3103c3961e7da993a3901a44b8ddc6a39e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.689ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}3x_{1}+5x_{2}=4\\x_{1}+2x_{2}=1\end{cases}}}"></span></dd></dl> <p>方程组中有两个未知数。以<a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a>表示，这个方程组可以记录为： </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{bmatrix}3&5\\1&2\end{bmatrix}}\cdot {\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}={\begin{bmatrix}4\\1\end{bmatrix}}}"> <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>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}3&5\\1&2\end{bmatrix}}\cdot {\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}={\begin{bmatrix}4\\1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7370e6a299ba2d62adfcff521973bd527787c3c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.591ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}3&5\\1&2\end{bmatrix}}\cdot {\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}={\begin{bmatrix}4\\1\end{bmatrix}}}"></span></dd></dl> <p>这个线性方程组有一组解：<span class="mwe-math-element"><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}=3,\,x_{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}=3,\,x_{2}=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/366697b874d0921a0886a6159c24834829eda704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.519ex; height:2.509ex;" alt="{\displaystyle x_{1}=3,\,x_{2}=-1}"></span>。可以直接验证： </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{cases}3\times 3+5\times (-1)=9-5=4\\3+2\times (-1)=3-2=1\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> <mn>3</mn> <mo>×</mo> <mn>3</mn> <mo>+</mo> <mn>5</mn> <mo>×</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>9</mn> <mo>−</mo> <mn>5</mn> <mo>=</mo> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mo>×</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}3\times 3+5\times (-1)=9-5=4\\3+2\times (-1)=3-2=1\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1431fe39c17fafce4312ff172289fe2643fe6679" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.808ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}3\times 3+5\times (-1)=9-5=4\\3+2\times (-1)=3-2=1\end{cases}}}"></span></dd></dl> <p>可以证明，这组解也是方程组唯一的解。 </p><p>不是所有的线性方程组都有解。以下是一个没有解的例子： </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{cases}x_{1}+x_{2}=2\\2x_{1}+2x_{2}=1\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> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}x_{1}+x_{2}=2\\2x_{1}+2x_{2}=1\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b75b23e41449a513cb425ef526b8a07b2c3635ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.689ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}x_{1}+x_{2}=2\\2x_{1}+2x_{2}=1\end{cases}}}"></span></dd></dl> <p>显然，如果有<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span>和<span class="mwe-math-element"><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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}"></span>满足了第一行的式子的话，它们的和等于2。而第二行则要求它们的和等于0.5，这不可能。 </p><p>也有的线性方程组有不止一组解。例如： </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{cases}x_{1}+x_{2}=2\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> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}x_{1}+x_{2}=2\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3abd4a6101a12391592b707a79202ab2f51a3e8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.783ex; height:2.843ex;" alt="{\displaystyle {\begin{cases}x_{1}+x_{2}=2\end{cases}}}"></span></dd></dl> <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}=1,\,x_{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}=1,\,x_{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf070387e139422784e645546bc645f9fd694c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.711ex; height:2.509ex;" alt="{\displaystyle x_{1}=1,\,x_{2}=1}"></span>是一组解，而<span class="mwe-math-element"><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}=3,\,x_{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}=3,\,x_{2}=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/366697b874d0921a0886a6159c24834829eda704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.519ex; height:2.509ex;" alt="{\displaystyle x_{1}=3,\,x_{2}=-1}"></span>也是一组解。事实上，解的个数有无限个。 </p> <div class="mw-heading mw-heading2"><h2 id="线性方程组的解"><span id=".E7.BA.BF.E6.80.A7.E6.96.B9.E7.A8.8B.E7.BB.84.E7.9A.84.E8.A7.A3"></span>线性方程组的解</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84&action=edit&section=2" title="编辑章节：线性方程组的解"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Intersecting_Lines.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Intersecting_Lines.svg/220px-Intersecting_Lines.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Intersecting_Lines.svg/330px-Intersecting_Lines.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Intersecting_Lines.svg/440px-Intersecting_Lines.svg.png 2x" data-file-width="500" data-file-height="500" /></a><figcaption>方程组的解是所有直线的公共点</figcaption></figure> <p>如果有一组数<span class="mwe-math-element"><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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2},\cdots ,x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6041920d47de448059e04bb211ff24e2fbe8e4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.528ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2},\cdots ,x_{n}}"></span>使得方程组的等号都成立，那么这组数就叫做方程组的解。一个线性方程组的所有的解的<a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a>会被简称为<b>解集</b>。根据解的存在情况，线性方程组可以分为三类： </p> <ul><li>有唯一解的恰定方程组，</li> <li>解不存在的超定方程组，</li> <li>有无穷多解的欠定方程组（也被通俗地称为不定方程组）。</li></ul> <div class="mw-heading mw-heading3"><h3 id="几何解释"><span id=".E5.87.A0.E4.BD.95.E8.A7.A3.E9.87.8A"></span>几何解释</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84&action=edit&section=3" title="编辑章节：几何解释"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>当未知数只有两个（<i>x</i>和<i>y</i>）的时候，方程组里面的每一个方程可以看成<b>Oxy</b><a href="/wiki/%E5%B9%B3%E9%9D%A2_(%E6%95%B0%E5%AD%A6)" title="平面 (数学)">平面</a>（正交直角坐标系）上的一条<a href="/wiki/%E7%9B%B4%E7%BA%BF" title="直线">直线</a>的方程。直线上的<a href="/wiki/%E7%82%B9" title="点">点</a>的坐标就是满足这个方程的一组数。从这个角度看来，方程组的解就是所有这种直线的<a href="/wiki/%E4%BA%A4%E9%BB%9E" title="交點">公共点</a>。而若干条直线的公共部分要么是一条直线，要么是一个点，要么是<a href="/wiki/%E7%A9%BA%E9%9B%86" title="空集">空集</a>，因此对应的，线性方程组的解要么有无穷个，要么恰好有一个，要么不存在。 </p><p>如果未知数有三个，那么每一个方程则代表了<a href="/wiki/%E4%B8%89%E7%BB%B4%E7%A9%BA%E9%97%B4" class="mw-redirect" title="三维空间">三维空间</a>里面的一个平面，而方程组的解集也就是一些平面的共同部分。所有解的集合可以对应一个平面，一条直线，一个点或空集。 </p><p>这个问题的一般情况可以从线性空间的角度去分析，即我们可以将线性方程组的求解问题看成向量<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }"></span>在矩阵<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span>所张成的<a href="/wiki/%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4" class="mw-redirect" title="线性空间">线性空间</a>里面的投影的问题。未知数的个数如果是一般的<i>n</i>个的话，可以想象每个方程代表了<a href="/wiki/N%E7%B6%AD%E7%A9%BA%E9%96%93" class="mw-redirect" title="N維空間"><b>n</b>维空间</a>里面的一个<a href="/wiki/%E8%B6%85%E5%B9%B3%E9%9D%A2" title="超平面">超平面</a>。而方程组的解就是所有超平面的公共点。 </p> <div class="mw-heading mw-heading3"><h3 id="齐次线性方程组"><span id=".E9.BD.90.E6.AC.A1.E7.BA.BF.E6.80.A7.E6.96.B9.E7.A8.8B.E7.BB.84"></span>齐次线性方程组</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84&action=edit&section=4" title="编辑章节：齐次线性方程组"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>齐次的线性方程组是指向量<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} =\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cdf97ca5b695e4defc6395e07c3148e7e7d1af0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.92ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} =\mathbf {0} }"></span>的情况。这时候方程变成： </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} \mathbf {x} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \mathbf {x} =\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f36f30ffbf41d69c696c67285e7ff385cbbf311" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.866ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} \mathbf {x} =\mathbf {0} }"></span></center> <p>这个方程肯定会有一组解：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecef8f00bf3524507cbbde1aec694e2237c9a0d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.846ex; height:2.176ex;" alt="{\displaystyle \mathbf {x} =\mathbf {0} }"></span>。实际上，方程的解就是矩阵<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span>对应的<a href="/wiki/%E7%BA%BF%E6%80%A7%E5%8F%98%E6%8D%A2" class="mw-redirect" title="线性变换">线性变换</a>的<a href="/wiki/%E9%9B%B6%E7%A9%BA%E9%97%B4" title="零空间">零空间</a>。一般来说，当方程的个数小于未知数的个数时，方程组会有除<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecef8f00bf3524507cbbde1aec694e2237c9a0d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.846ex; height:2.176ex;" alt="{\displaystyle \mathbf {x} =\mathbf {0} }"></span>以外的解。当方程组个数变多时，则要看其中“有效”的方程的个数。有时候某一个方程可以表示成另外几个方程的线性组合。比如方程组： </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}3x_{1}+x_{2}+2x_{3}=0\\x_{1}-x_{2}+4x_{3}=0\\2x_{1}+3x_{3}=0\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> <mn>3</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>4</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}3x_{1}+x_{2}+2x_{3}=0\\x_{1}-x_{2}+4x_{3}=0\\2x_{1}+3x_{3}=0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4202381283bdd1e437ddfe6246698071e7c52b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:22.236ex; height:8.509ex;" alt="{\displaystyle {\begin{cases}3x_{1}+x_{2}+2x_{3}=0\\x_{1}-x_{2}+4x_{3}=0\\2x_{1}+3x_{3}=0\end{cases}}}"></span></center> <p>之中，第三个方程就可以表示为前两个方程的线性组合： </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x_{1}+3x_{3}={\frac {1}{2}}(3x_{1}+x_{2}+2x_{3})+{\frac {1}{2}}(x_{1}-x_{2}+4x_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>4</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x_{1}+3x_{3}={\frac {1}{2}}(3x_{1}+x_{2}+2x_{3})+{\frac {1}{2}}(x_{1}-x_{2}+4x_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed1a8dc8443a1a03876f540d7dfd812596fd944" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:52.64ex; height:5.176ex;" alt="{\displaystyle 2x_{1}+3x_{3}={\frac {1}{2}}(3x_{1}+x_{2}+2x_{3})+{\frac {1}{2}}(x_{1}-x_{2}+4x_{3})}"></span></center> <p>这时第三个方程组就可以不必考虑了。用线性代数的词汇表达，“有效”的方程的个数就是矩阵<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span>中线性无关的行向量的个数，或者说行向量线性张成的空间的维数。这个数也被称为<a href="/wiki/%E7%9F%A9%E9%98%B5%E7%9A%84%E7%A7%A9" class="mw-redirect" title="矩阵的秩">矩阵的秩</a>。当矩阵<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span>的秩<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>小于未知数的个数<span class="mwe-math-element"><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>时，方程组的解会有无穷多个，构成一个<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86c0aa8d9c52b096540edd6e6a91eb8f790a8b7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.284ex; height:2.176ex;" alt="{\displaystyle n-r}"></span>维的线性空间。而当<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>等于未知数的个数<span class="mwe-math-element"><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>时，方程组有唯一零解，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>不可能大于<span class="mwe-math-element"><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> <div class="mw-heading mw-heading3"><h3 id="松弛求解"><span id=".E6.9D.BE.E5.BC.9B.E6.B1.82.E8.A7.A3"></span>松弛求解</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84&action=edit&section=5" title="编辑章节：松弛求解"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>在实验数据处理和曲线拟合问题中，求解超定方程组非常普遍。这时常常需要退一步，将线性方程组的求解问题改变为求最小误差的问题。形象的说，就是在无法完全满足给定的这些条件的情况下，求一个最接近的解。比较常用的方法是<a href="/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98%E6%B3%95" title="最小二乘法">最小二乘法</a>。最小二乘法求解超定问题等价于一个<a href="/wiki/%E4%BC%98%E5%8C%96%E9%97%AE%E9%A2%98" class="mw-redirect" title="优化问题">优化问题</a>，或者说最小值问题，即，在不存在<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"></span>使得<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} -\mathbf {A} \mathbf {x} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} -\mathbf {A} \mathbf {x} =\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85d0ada9af064e24dbc87b89abdacae3a2a84a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.191ex; height:2.343ex;" alt="{\displaystyle \mathbf {b} -\mathbf {A} \mathbf {x} =\mathbf {0} }"></span>的情况下，我们试图找到这样的<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"></span>使得<span class="mwe-math-element"><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|\mathbf {b} -\mathbf {A} \mathbf {x} \right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\mathbf {b} -\mathbf {A} \mathbf {x} \right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a2283c6fd868a4aebef25f05199f5c1f1a26f39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.05ex; height:2.843ex;" alt="{\displaystyle \left|\mathbf {b} -\mathbf {A} \mathbf {x} \right|}"></span>最小，其中<span class="mwe-math-element"><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|\cdot \right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mo>⋅</mo> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\cdot \right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8602a75a86dbb13d4c93cb88b01db106d37b4f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.941ex; height:2.843ex;" alt="{\displaystyle \left|\cdot \right|}"></span>表示<a href="/wiki/%E8%8C%83%E6%95%B0" title="范数">范数</a>。 </p> <div class="mw-heading mw-heading2"><h2 id="求解"><span id=".E6.B1.82.E8.A7.A3"></span>求解</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84&action=edit&section=6" title="编辑章节：求解"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="克莱姆法则"><span id=".E5.85.8B.E8.8E.B1.E5.A7.86.E6.B3.95.E5.88.99"></span>克莱姆法则</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84&action=edit&section=7" title="编辑章节：克莱姆法则"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>基于线性方程组的解空间理论，线性方程组有唯一解当且仅当有效方程数等于未知数的个数。这时，可以运用各种方法具体求出唯一存在的解。<a href="/wiki/%E5%85%8B%E8%90%8A%E5%A7%86%E6%B3%95%E5%89%87" title="克萊姆法則">克萊姆法則</a>是一种求解<b>线性方程组</b>的方法，大多数<a href="/wiki/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="线性代数">线性代数</a>教材都会提到。例如对于如下的线性方程组： </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}a_{1}x+b_{1}y=c_{1}\\a_{2}x+b_{2}y=c_{2}\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> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>y</mi> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>y</mi> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}a_{1}x+b_{1}y=c_{1}\\a_{2}x+b_{2}y=c_{2}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4859fad24e46ea0da07f2438c236639c51064d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.316ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}a_{1}x+b_{1}y=c_{1}\\a_{2}x+b_{2}y=c_{2}\end{cases}}}"></span></center> <p>运用克莱姆法则，这个方程组的解可以如下： </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {D_{x}}{D}},\qquad y={\frac {D_{y}}{D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>D</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>D</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {D_{x}}{D}},\qquad y={\frac {D_{y}}{D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63c110de903a8ef7b01c0e45cb300e23eac4dc1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.104ex; height:5.509ex;" alt="{\displaystyle x={\frac {D_{x}}{D}},\qquad y={\frac {D_{y}}{D}}}"></span></center> <p>其中，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{x},D_{y},D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{x},D_{y},D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec4a8974a60cdd8d132b4de190c310f5a224997" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.062ex; height:2.843ex;" alt="{\displaystyle D_{x},D_{y},D}"></span>分别是如下三个<a href="/wiki/%E8%A1%8C%E5%88%97%E5%BC%8F" title="行列式">行列式</a>： </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=\left|{\begin{matrix}a_{1}&b_{1}\\a_{2}&b_{2}\end{matrix}}\right|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=\left|{\begin{matrix}a_{1}&b_{1}\\a_{2}&b_{2}\end{matrix}}\right|,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f4ec6c0397861aa2f6ca76bee6cfa56e5631bc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.76ex; height:6.176ex;" alt="{\displaystyle D=\left|{\begin{matrix}a_{1}&b_{1}\\a_{2}&b_{2}\end{matrix}}\right|,}"></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 D_{x}=\left|{\begin{matrix}c_{1}&b_{1}\\c_{2}&b_{2}\end{matrix}}\right|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{x}=\left|{\begin{matrix}c_{1}&b_{1}\\c_{2}&b_{2}\end{matrix}}\right|,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a039c2b13f330238c098e5c75347d67d76a5a8af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.71ex; height:6.176ex;" alt="{\displaystyle D_{x}=\left|{\begin{matrix}c_{1}&b_{1}\\c_{2}&b_{2}\end{matrix}}\right|,}"></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 D_{y}=\left|{\begin{matrix}a_{1}&c_{1}\\a_{2}&c_{2}\end{matrix}}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{y}=\left|{\begin{matrix}a_{1}&c_{1}\\a_{2}&c_{2}\end{matrix}}\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96dc68abaf2b212a2982b2c89ab01f26caf79b60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.785ex; height:6.176ex;" alt="{\displaystyle D_{y}=\left|{\begin{matrix}a_{1}&c_{1}\\a_{2}&c_{2}\end{matrix}}\right|}"></span></center> <p>对于更一般的情况： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} \mathbf {x} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \mathbf {x} =\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1688bdcfe95659c92f6452378ee805da3f796c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.014ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} \mathbf {x} =\mathbf {b} }"></span></dd></dl> <p>解可以由同样的公式给出： </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}x_{1}={\frac {D_{1}}{D}}\\x_{2}={\frac {D_{2}}{D}}\\\vdots \qquad \vdots \\x_{n}={\frac {D_{n}}{D}}\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> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>D</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>D</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> <mspace width="2em" /> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>D</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}x_{1}={\frac {D_{1}}{D}}\\x_{2}={\frac {D_{2}}{D}}\\\vdots \qquad \vdots \\x_{n}={\frac {D_{n}}{D}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58f16be7c9e4dea134580ab0e56def98c37e5407" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.043ex; margin-bottom: -0.295ex; width:11.626ex; height:15.843ex;" alt="{\displaystyle {\begin{cases}x_{1}={\frac {D_{1}}{D}}\\x_{2}={\frac {D_{2}}{D}}\\\vdots \qquad \vdots \\x_{n}={\frac {D_{n}}{D}}\end{cases}}}"></span></center> <p>其中的 </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 D=\det(\mathbf {A} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=\det(\mathbf {A} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c98c8dcbc5159347171244cb12ef8286b4c379" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.081ex; height:2.843ex;" alt="{\displaystyle D=\det(\mathbf {A} )}"></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 \forall 1\leqslant i\leqslant n,\,\,D_{i}=\det(\mathbf {A_{i}} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mn>1</mn> <mo>⩽</mo> <mi>i</mi> <mo>⩽</mo> <mi>n</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall 1\leqslant i\leqslant n,\,\,D_{i}=\det(\mathbf {A_{i}} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45f439688b9bc6f012fd8746496f39cfafcee3d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.295ex; height:2.843ex;" alt="{\displaystyle \forall 1\leqslant i\leqslant n,\,\,D_{i}=\det(\mathbf {A_{i}} )}"></span></dd></dl> <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 \mathbf {A_{i}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A_{i}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75f909e777b93b8297e8576aa84d6ee4a17b8669" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.776ex; height:2.509ex;" alt="{\displaystyle \mathbf {A_{i}} }"></span>是将矩阵<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span>的第<i>i</i>纵列换成向量<b>b</b>之后得到的矩阵。 </p><p>可以看出，这些表达式只有在<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=\det(\mathbf {A} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=\det(\mathbf {A} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c98c8dcbc5159347171244cb12ef8286b4c379" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.081ex; height:2.843ex;" alt="{\displaystyle D=\det(\mathbf {A} )}"></span>存在并且不等于0的时候才是有意义的，这点只有在有效方程数等于未知数的个数的时候才能得到保证。 </p> <div class="mw-heading mw-heading3"><h3 id="数值方法"><span id=".E6.95.B0.E5.80.BC.E6.96.B9.E6.B3.95"></span>数值方法</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84&action=edit&section=8" title="编辑章节：数值方法"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>在实际运算中，当矩阵的维数较高时，计算<a href="/wiki/%E8%A1%8C%E5%88%97%E5%BC%8F" title="行列式">行列式</a>是非常繁複的。也就是说，计算行列式的<a href="/wiki/%E8%AE%A1%E7%AE%97%E5%A4%8D%E6%9D%82%E5%BA%A6" class="mw-redirect" title="计算复杂度">计算复杂度</a>随维数的增长非常快，对于一个<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span>的矩阵，用初等的方法计算其行列式，需要的计算时间是<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n!)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n!)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12921c489714d475a454bd39ef644d4334d97113" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.624ex; height:2.843ex;" alt="{\displaystyle O(n!)}"></span>（n的阶乘）。因此，克莱姆法则在現實問題求解幾乎不會被采用，而其重要性在于證明某些線性問題的解答會自然而然是整數解答，因而存在有效率的解法。換言之，克萊姆法則的重要性是在於理論證明的應用，而非問題的實際求解。 </p><p>经典的求解线性方程组的方法一般分为两类：直接法和迭代法。前者例如<a href="/wiki/%E9%AB%98%E6%96%AF%E6%B6%88%E5%8E%BB%E6%B3%95" title="高斯消去法">高斯消去法</a>, <a href="/wiki/LU%E5%88%86%E8%A7%A3" title="LU分解">LU分解</a>等，后者的例子包括<a href="/wiki/%E5%85%B1%E8%BD%AD%E6%A2%AF%E5%BA%A6%E6%B3%95" title="共轭梯度法">共轭梯度法</a>等。这些方法的计算复杂度在可以接受的范围内，因此被广泛采用。例如，高斯消去法的复杂度为<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n^{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b04f5c5cfea38f43406d9442387ad28555e2609" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.032ex; height:3.176ex;" alt="{\displaystyle O(n^{3})}"></span> 一般来说，直接法对于阶数比较低的方程组（少于20000至30000个未知数）比较有效；而后者对于比较大的方程组更有效。在实际计算中，几十万甚至几百万个未知数的方程组并不少见。在这些情况下，迭代法有无可比拟的优势。另外，使用迭代法可以根据不同的精度要求选择终止时间，因此比较灵活。在问题特别大的时候，计算机内存可能无法容纳被操作的矩阵，这给直接法带来很大的挑战。而对于迭代法，则可以将矩阵的某一部分读入内存进行操作，然后再操作另外部分。 </p> <div class="mw-heading mw-heading2"><h2 id="应用"><span id=".E5.BA.94.E7.94.A8"></span>应用</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84&action=edit&section=9" title="编辑章节：应用"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>现实中的问题大多数是连续的，例如工程中求解结构受力后的变形，<a href="/wiki/%E7%A9%BA%E6%B0%94%E5%8A%A8%E5%8A%9B%E5%AD%A6" title="空气动力学">空气动力学</a>中计算机翼周围的流场，气象预报中计算大气的流动。这些现象大多是用若干个<a href="/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="微分方程">微分方程</a>描述。用<a href="/wiki/%E6%95%B0%E5%80%BC%E6%96%B9%E6%B3%95" class="mw-redirect" title="数值方法">数值方法</a>求解微分方程（组），不论是<a href="/w/index.php?title=%E5%B7%AE%E5%88%86%E6%96%B9%E6%B3%95&action=edit&redlink=1" class="new" title="差分方法（页面不存在）">差分方法</a>还是<a href="/wiki/%E6%9C%89%E9%99%90%E5%85%83%E6%96%B9%E6%B3%95" class="mw-redirect" title="有限元方法">有限元方法</a>，通常都是通过对微分方程（连续的问题，未知数的维数是无限的）进行离散，得到线性方程组（离散问题，因为未知数的维数是有限的）。因此线性方程组的求解在科学与工程中的应用非常广泛。 </p><p>许多具体的应用会得到结构比较特别的线性方程组，比如用差分方法和有限元方法离散微分方程后通常会得到三对角或五对角的方程组，网络问题有时会得到对称的线性方程组（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} =\mathbf {A} ^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} =\mathbf {A} ^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05f74d19a2467e53b9b719ae36ab49b6314036cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.527ex; height:2.676ex;" alt="{\displaystyle \mathbf {A} =\mathbf {A} ^{T}}"></span>），因此除了通用的线性方程组求解器，在一些专业领域，研究人员们也开发了适用于特定问题的求解器，比如适用于<a href="/wiki/%E7%A8%80%E7%96%8F%E7%9F%A9%E9%98%B5" title="稀疏矩阵">稀疏矩阵</a>的求解器，适用于三对角矩阵的求解器，适用于对称矩阵的求解器等。 </p> <div class="mw-heading mw-heading2"><h2 id="相关软件"><span id=".E7.9B.B8.E5.85.B3.E8.BD.AF.E4.BB.B6"></span>相关软件</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84&action=edit&section=10" title="编辑章节：相关软件"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>由于线性方程组的求解是一个非常普遍的问题，在多年的科学与工程实践中，科学家和工程师们积累很多高效率的线性方程组求解器，例如：LAPACK、BLAS等。这些软件中，许多可以可以在<a rel="nofollow" class="external text" href="http://www.netlib.org">NetLib</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20071024041352/http://www.netlib.org/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）免费获得。LAPACK和BLAS在大多数<a href="/wiki/Linux" title="Linux">Linux</a>的发行版本中都已经预装。目前LAPACK有<a href="/wiki/Fortran" title="Fortran">Fortran</a>（包括90和77版本）、C、<a href="/wiki/C%2B%2B" title="C++">C++</a>等几个语言的版本。利用LAPACK和BLAS中的子程序，<a rel="nofollow" class="external text" href="http://www.mathworks.com">Matlab</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20210505033254/http://www.mathworks.com/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）对这些线性方程组求解器进行封装。用户不需要选择求解器的类型和问题的类型，<a href="/wiki/Matlab" class="mw-redirect" title="Matlab">Matlab</a>可以根据对矩阵的分析自动选择合适的求解器。 </p> <div class="mw-heading mw-heading2"><h2 id="其他方法与软件"><span id=".E5.85.B6.E4.BB.96.E6.96.B9.E6.B3.95.E4.B8.8E.E8.BD.AF.E4.BB.B6"></span>其他方法与软件</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84&action=edit&section=11" title="编辑章节：其他方法与软件"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>上面讲的是线性方程组的数值解法。对于比较小的线性方程组，求得符号解是可能的。常用的软件有Mathematica, Maple等。在某些领域的研究中，这种需要并且可能求符号解（精确解）的情况偶尔会遇到。未知数的个数一般限制在几十个左右。显然，符号解在对于实际中遇到的有几百万个未知数的问题是无能为力的，比如，大型结构，天气预报，湍流模拟等问题中得到的线性方程组。 </p> <div class="mw-heading mw-heading2"><h2 id="参见"><span id=".E5.8F.82.E8.A7.81"></span>参见</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84&action=edit&section=12" title="编辑章节：参见"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E6%96%B9%E7%A8%8B" title="方程">方程</a></li> <li><a href="/wiki/%E4%B8%8D%E5%AE%9A%E6%96%B9%E7%A8%8B" class="mw-redirect" title="不定方程">不定方程</a></li> <li><a href="/wiki/%E9%B8%A1%E5%85%94%E5%90%8C%E7%AC%BC" title="鸡兔同笼">鸡兔同笼</a></li> <li><a href="/wiki/%E5%AF%B9%E8%A7%92%E5%8C%96" class="mw-redirect" title="对角化">对角化</a></li> <li><a href="/wiki/%E9%80%86%E7%9F%A9%E9%98%B5" title="逆矩阵">逆矩阵</a></li> <li><a href="/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="微分方程">微分方程</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="参考文献"><span id=".E5.8F.82.E8.80.83.E6.96.87.E7.8C.AE"></span>参考文献</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84&action=edit&section=13" title="编辑章节：参考文献"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small"> <ul><li><a rel="nofollow" class="external free" href="https://web.archive.org/web/20070208024707/http://www.okc.cc.ok.us/maustin/Cramers_Rule/Cramer%27s%20Rule.htm">https://web.archive.org/web/20070208024707/http://www.okc.cc.ok.us/maustin/Cramers_Rule/Cramer%27s%20Rule.htm</a></li></ul> <p><br /> </p> <ul><li><a rel="nofollow" class="external free" href="http://www.netlib.org/lapack/">http://www.netlib.org/lapack/</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20110224085358/http://www.netlib.org/lapack/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li></ul> <p><br /> </p> <ul><li><a rel="nofollow" class="external free" href="http://www.mathworks.com">http://www.mathworks.com</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20210505033254/http://www.mathworks.com/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="外部連結"><span id=".E5.A4.96.E9.83.A8.E9.80.A3.E7.B5.90"></span>外部連結</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84&action=edit&section=14" title="编辑章节：外部連結"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="英語">（英文）</span><a rel="nofollow" class="external text" href="http://www.mathway.com/">Mathway</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20210325145233/http://www.mathway.com/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>），使用電腦解決絕大部分的數學問題。</li> <li><span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="英語">（英文）</span><a rel="nofollow" class="external text" href="http://www.idomaths.com/simeq.php">Simultaneous Linear Equations Calculator</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20201125073035/http://www.idomaths.com/simeq.php">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>），一個簡單的线性<a href="/wiki/%E6%96%B9%E7%A8%8B%E7%BB%84" title="方程组">方程组</a>求解器</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><style data-mw-deduplicate="TemplateStyles:r84261037">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{text-align:center;padding-left:1em;padding-right:1em}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{border-color:#fdfdfd}.mw-parser-output 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href="/wiki/Template:%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%E7%9A%84%E7%9B%B8%E5%85%B3%E6%A6%82%E5%BF%B5" title="Template:线性代数的相关概念"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%E7%9A%84%E7%9B%B8%E5%85%B3%E6%A6%82%E5%BF%B5&action=edit&redlink=1" class="new" title="Template talk:线性代数的相关概念（页面不存在）"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%E7%9A%84%E7%9B%B8%E5%85%B3%E6%A6%82%E5%BF%B5" title="Special:编辑页面/Template:线性代数的相关概念"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="线性代数的相关概念" style="font-size:110%;margin:0 5em"><a href="/wiki/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="线性代数">线性代数</a>的相关概念</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">重要概念</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E6%A0%87%E9%87%8F_(%E6%95%B0%E5%AD%A6)" title="标量 (数学)">标量</a></li> <li><a href="/wiki/%E5%90%91%E9%87%8F" title="向量">向量</a></li> <li><a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间">向量空间</a></li> <li><a href="/wiki/%E7%BA%BF%E6%80%A7%E5%AD%90%E7%A9%BA%E9%97%B4" title="线性子空间">向量子空间</a></li></ul> <ul><li><a href="/wiki/%E7%BA%BF%E6%80%A7%E7%94%9F%E6%88%90%E7%A9%BA%E9%97%B4" title="线性生成空间">线性生成空间</a></li> <li><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="线性映射">线性映射</a></li> <li><a href="/wiki/%E6%8A%95%E5%BD%B1" class="mw-disambig" title="投影">投影</a></li> <li><a href="/wiki/%E7%B7%9A%E6%80%A7%E7%84%A1%E9%97%9C" title="線性無關">線性無關</a></li> <li><a href="/wiki/%E7%BA%BF%E6%80%A7%E7%BB%84%E5%90%88" title="线性组合">线性组合</a></li></ul> <ul><li><a href="/wiki/%E5%9F%BA_(%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8)" title="基 (線性代數)">基</a></li> <li><a href="/wiki/%E6%A8%99%E8%A8%98_(%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8)" title="標記 (線性代數)">標記</a></li> <li><a href="/wiki/%E5%88%97%E7%A9%BA%E9%97%B4" class="mw-redirect" title="列空间">列空间</a></li> <li><a href="/wiki/%E8%A1%8C%E7%A9%BA%E9%97%B4" class="mw-redirect" title="行空间">行空间</a></li> <li><a href="/wiki/%E9%9B%B6%E7%A9%BA%E9%97%B4" title="零空间">零空间</a></li> <li><a href="/wiki/%E5%AF%B9%E5%81%B6%E7%A9%BA%E9%97%B4" title="对偶空间">对偶空间</a></li> <li><a href="/wiki/%E6%AD%A3%E4%BA%A4" title="正交">正交</a></li> <li><a href="/wiki/%E7%89%B9%E5%BE%81%E5%80%BC" class="mw-redirect" title="特征值">特征值</a></li> <li><a href="/wiki/%E7%89%B9%E5%BE%81%E5%90%91%E9%87%8F" class="mw-redirect" title="特征向量">特征向量</a></li></ul> <ul><li><a href="/wiki/%E7%82%B9%E7%A7%AF" title="点积">数量积</a></li> <li><a href="/wiki/%E5%86%85%E7%A7%AF%E7%A9%BA%E9%97%B4" title="内积空间">内积空间</a></li> <li><a href="/wiki/%E7%82%B9%E4%B9%98" class="mw-redirect" title="点乘">点乘</a></li> <li><a href="/wiki/%E8%BD%89%E7%BD%AE" class="mw-redirect" title="轉置">轉置</a></li> <li><a href="/wiki/%E6%A0%BC%E6%8B%89%E5%A7%86-%E6%96%BD%E5%AF%86%E7%89%B9%E6%AD%A3%E4%BA%A4%E5%8C%96" title="格拉姆-施密特正交化">格拉姆-施密特正交化</a></li> <li><a class="mw-selflink selflink">线性方程组</a></li> <li><a href="/wiki/%E5%85%8B%E8%90%8A%E5%A7%86%E6%B3%95%E5%89%87" title="克萊姆法則">克萊姆法則</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">矩阵</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a></li> <li><a href="/wiki/%E7%9F%A9%E9%99%A3%E4%B9%98%E6%B3%95" title="矩陣乘法">矩陣乘法</a></li> <li><a href="/wiki/%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%A3" title="矩阵分解">矩阵分解</a></li> <li><a href="/wiki/%E8%A1%8C%E5%88%97%E5%BC%8F" title="行列式">行列式</a></li> <li><a href="/wiki/%E5%AD%90%E5%BC%8F%E5%92%8C%E4%BD%99%E5%AD%90%E5%BC%8F" title="子式和余子式">子式和余子式</a></li> <li><a href="/wiki/%E7%A7%A9_(%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0)" title="秩 (线性代数)">矩阵的秩</a></li> <li><a href="/wiki/%E5%85%8B%E8%90%8A%E5%A7%86%E6%B3%95%E5%89%87" title="克萊姆法則">克萊姆法則</a></li> <li><a href="/wiki/%E9%80%86%E7%9F%A9%E9%98%B5" title="逆矩阵">逆矩阵</a></li> <li><a href="/wiki/%E9%AB%98%E6%96%AF%E6%B6%88%E5%8E%BB%E6%B3%95" title="高斯消去法">高斯消去法</a></li> <li><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="线性映射">线性变换</a></li> <li><a href="/wiki/%E5%88%86%E5%A1%8A%E7%9F%A9%E9%99%A3" title="分塊矩陣">分块矩阵</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%95%B0%E5%80%BC%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="数值线性代数">数值线性代数</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E6%B5%AE%E7%82%B9%E6%95%B0%E8%BF%90%E7%AE%97" title="浮点数运算">浮点数</a></li> <li><a href="/wiki/%E6%95%B0%E5%80%BC%E7%A8%B3%E5%AE%9A%E6%80%A7" title="数值稳定性">数值稳定性</a></li> <li><a href="/wiki/BLAS" title="BLAS">基础线性代数程序集</a></li> <li><a href="/wiki/%E7%A8%80%E7%96%8F%E7%9F%A9%E9%98%B5" title="稀疏矩阵">稀疏矩阵</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84261037"></div><div role="navigation" class="navbox authority-control" 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/Help:%E8%A7%84%E8%8C%83%E6%8E%A7%E5%88%B6" title="Help:规范控制">规范控制数据库</a>：各地 <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11203#identifiers" title="編輯維基數據鏈接"><img alt="編輯維基數據鏈接" 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