线性映射 - 维基百科，自由的百科全书

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="zh" dir="ltr"> <head> <meta charset="UTF-8"> <title>线性映射 - 维基百科，自由的百科全书</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )zhwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"zh", "wgMonthNames":["","1月","2月","3月","4月","5月","6月","7月","8月","9月","10月","11月","12月"],"wgRequestId":"b4732398-3285-4b18-9999-8cfddb8c1c8d","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"线性映射","wgTitle":"线性映射","wgCurRevisionId":84918282,"wgRevisionId":84918282,"wgArticleId":564925,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["含有英語的條目","自2016年6月扩充中的条目","所有扩充中的条目","拒绝当选首页新条目推荐栏目的条目","使用小型訊息框的頁面","使用ISBN魔术链接的页面","使用过时的math标签格式的页面","函数","抽象代数","線性代數","线性算子"],"wgPageViewLanguage":"zh","wgPageContentLanguage":"zh","wgPageContentModel":"wikitext","wgRelevantPageName":"线性映射","wgRelevantArticleId":564925,"wgUserVariant":"zh","wgIsProbablyEditable":true, "wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"zh","pageLanguageDir":"ltr","pageVariantFallbacks":["zh-hans","zh-hant","zh-cn","zh-tw","zh-hk","zh-sg","zh-mo","zh-my"]},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q207643","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"], "GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.gadget.large-font":"ready","ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.edit0", "ext.gadget.WikiMiniAtlas","ext.gadget.UnihanTooltips","ext.gadget.Difflink","ext.gadget.pseudonamespace-UI","ext.gadget.SpecialWikitext","ext.gadget.switcher","ext.gadget.VariantAlly","ext.gadget.AdvancedSiteNotices","ext.gadget.hideConversionTab","ext.gadget.internalLinkHelper-altcolor","ext.gadget.noteTA","ext.gadget.NavFrame","ext.gadget.collapsibleTables","ext.gadget.scrollUpButton","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=zh&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=zh&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=zh&modules=ext.gadget.large-font&only=styles&skin=vector-2022"> <link rel="stylesheet" href="/w/load.php?lang=zh&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="线性映射 - 维基百科，自由的百科全书"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//zh.m.wikipedia.org/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84"> <link rel="alternate" type="application/x-wiki" title="编辑本页" href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (zh)"> <link rel="EditURI" type="application/rsd+xml" href="//zh.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://zh.wikipedia.org/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84"> <link rel="alternate" hreflang="zh" href="https://zh.wikipedia.org/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84"> <link rel="alternate" hreflang="zh-Hans" href="https://zh.wikipedia.org/zh-hans/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84"> <link rel="alternate" hreflang="zh-Hans-CN" href="https://zh.wikipedia.org/zh-cn/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84"> <link rel="alternate" hreflang="zh-Hans-MY" href="https://zh.wikipedia.org/zh-my/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84"> <link rel="alternate" hreflang="zh-Hans-SG" href="https://zh.wikipedia.org/zh-sg/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84"> <link rel="alternate" hreflang="zh-Hant" href="https://zh.wikipedia.org/zh-hant/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84"> <link rel="alternate" hreflang="zh-Hant-HK" href="https://zh.wikipedia.org/zh-hk/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84"> <link rel="alternate" hreflang="zh-Hant-MO" href="https://zh.wikipedia.org/zh-mo/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84"> <link rel="alternate" hreflang="zh-Hant-TW" href="https://zh.wikipedia.org/zh-tw/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84"> <link rel="alternate" hreflang="x-default" href="https://zh.wikipedia.org/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.zh"> <link rel="alternate" type="application/atom+xml" title="Wikipedia的Atom feed" href="/w/index.php?title=Special:%E6%9C%80%E8%BF%91%E6%9B%B4%E6%94%B9&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-线性映射 rootpage-线性映射 skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">跳转到内容</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="主菜单" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">主菜单</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">主菜单</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">隐藏</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> 导航 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Wikipedia:%E9%A6%96%E9%A1%B5" title="访问首页[z]" accesskey="z"><span>首页</span></a></li><li id="n-indexpage" class="mw-list-item"><a href="/wiki/Wikipedia:%E5%88%86%E7%B1%BB%E7%B4%A2%E5%BC%95" title="以分类索引搜寻中文维基百科"><span>分类索引</span></a></li><li id="n-Featured_content" class="mw-list-item"><a href="/wiki/Portal:%E7%89%B9%E8%89%B2%E5%85%A7%E5%AE%B9" title="查看中文维基百科的特色内容"><span>特色内容</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:%E6%96%B0%E8%81%9E%E5%8B%95%E6%85%8B" title="提供当前新闻事件的背景资料"><span>新闻动态</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:%E6%9C%80%E8%BF%91%E6%9B%B4%E6%94%B9" title="列出维基百科中的最近修改[r]" accesskey="r"><span>最近更改</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:%E9%9A%8F%E6%9C%BA%E9%A1%B5%E9%9D%A2" title="随机载入一个页面[x]" accesskey="x"><span>随机条目</span></a></li> </ul> </div> </div> <div id="p-help" class="vector-menu mw-portlet mw-portlet-help" > <div class="vector-menu-heading"> 帮助 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:%E7%9B%AE%E5%BD%95" title="寻求帮助"><span>帮助</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:%E7%A4%BE%E7%BE%A4%E9%A6%96%E9%A1%B5" title="关于本计划、你可以做什么、应该如何做"><span>维基社群</span></a></li><li id="n-policy" class="mw-list-item"><a href="/wiki/Wikipedia:%E6%96%B9%E9%87%9D%E8%88%87%E6%8C%87%E5%BC%95" title="查看维基百科的方针和指引"><span>方针与指引</span></a></li><li id="n-villagepump" class="mw-list-item"><a href="/wiki/Wikipedia:%E4%BA%92%E5%8A%A9%E5%AE%A2%E6%A0%88" title="参与维基百科社群的讨论"><span>互助客栈</span></a></li><li id="n-Information_desk" class="mw-list-item"><a href="/wiki/Wikipedia:%E7%9F%A5%E8%AF%86%E9%97%AE%E7%AD%94" title="解答任何与维基百科无关的问题的地方"><span>知识问答</span></a></li><li id="n-conversion" class="mw-list-item"><a href="/wiki/Wikipedia:%E5%AD%97%E8%AF%8D%E8%BD%AC%E6%8D%A2" title="提出字词转换请求"><span>字词转换</span></a></li><li id="n-IRC" class="mw-list-item"><a href="/wiki/Wikipedia:IRC%E8%81%8A%E5%A4%A9%E9%A2%91%E9%81%93"><span>IRC即时聊天</span></a></li><li id="n-contact" class="mw-list-item"><a href="/wiki/Wikipedia:%E8%81%94%E7%BB%9C%E6%88%91%E4%BB%AC" title="如何联络维基百科"><span>联络我们</span></a></li><li id="n-about" class="mw-list-item"><a href="/wiki/Wikipedia:%E5%85%B3%E4%BA%8E" title="查看维基百科的简介"><span>关于维基百科</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Wikipedia:%E9%A6%96%E9%A1%B5" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="维基百科" src="/static/images/mobile/copyright/wikipedia-wordmark-zh.svg" style="width: 6.5625em; height: 1.375em;"> <img class="mw-logo-tagline" alt="自由的百科全书" src="/static/images/mobile/copyright/wikipedia-tagline-zh.svg" width="103" height="14" style="width: 6.4375em; height: 0.875em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:%E6%90%9C%E7%B4%A2" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="搜索维基百科[f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>搜索</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="搜索维基百科" aria-label="搜索维基百科" autocapitalize="sentences" title="搜索维基百科[f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:搜索"> </div> <button class="cdx-button cdx-search-input__end-button">搜索</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="个人工具"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="外观"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="更改页面字体大小、宽度和颜色的外观" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="外观" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">外观</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?utm_source=donate&utm_medium=sidebar&utm_campaign=spontaneous&uselang=zh-hans" class=""><span>资助维基百科</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:%E5%88%9B%E5%BB%BA%E8%B4%A6%E6%88%B7&returnto=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="我们推荐您创建账号并登录，但这不是强制性的" class=""><span>创建账号</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" 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%98%A0%E5%B0%84" title="建议你登录，尽管并非必须。[o]" accesskey="o" class=""><span>登录</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="更多选项" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="个人工具" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">个人工具</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="用户菜单" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?utm_source=donate&utm_medium=sidebar&utm_campaign=spontaneous&uselang=zh-hans"><span>资助维基百科</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E5%88%9B%E5%BB%BA%E8%B4%A6%E6%88%B7&returnto=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="我们推荐您创建账号并登录，但这不是强制性的"><span 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%98%A0%E5%B0%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-正式定義" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#正式定義"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>正式定義</span> </div> </a> <button aria-controls="toc-正式定義-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>开关正式定義子章节</span> </button> <ul id="toc-正式定義-sublist" class="vector-toc-list"> <li id="toc-線性泛函" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#線性泛函"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>線性泛函</span> </div> </a> <ul id="toc-線性泛函-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-注意事項" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#注意事項"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>注意事項</span> </div> </a> <ul id="toc-注意事項-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-相關名詞" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#相關名詞"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>相關名詞</span> </div> </a> <ul id="toc-相關名詞-sublist" class="vector-toc-list"> </ul> </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">3</span> <span>例子</span> </div> </a> <ul id="toc-例子-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-矩陣" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#矩陣"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>矩陣</span> </div> </a> <button aria-controls="toc-矩陣-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>开关矩陣子章节</span> </button> <ul id="toc-矩陣-sublist" class="vector-toc-list"> <li id="toc-以矩陣表示線性映射" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#以矩陣表示線性映射"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>以矩陣表示線性映射</span> </div> </a> <ul id="toc-以矩陣表示線性映射-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-以線性映射表示矩陣" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#以線性映射表示矩陣"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>以線性映射表示矩陣</span> </div> </a> <ul id="toc-以線性映射表示矩陣-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-用泛性质做矩阵表示" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#用泛性质做矩阵表示"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>用泛性质做矩阵表示</span> </div> </a> <ul id="toc-用泛性质做矩阵表示-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-用矩陣表示線性映射的原因和好處" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#用矩陣表示線性映射的原因和好處"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>用矩陣表示線性映射的原因和好處</span> </div> </a> <ul id="toc-用矩陣表示線性映射的原因和好處-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-線性映射的矩陣的例子" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#線性映射的矩陣的例子"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>線性映射的矩陣的例子</span> </div> </a> <ul id="toc-線性映射的矩陣的例子-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-從給定線性映射構造新的線性映射" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#從給定線性映射構造新的線性映射"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>從給定線性映射構造新的線性映射</span> </div> </a> <ul id="toc-從給定線性映射構造新的線性映射-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-自同態線性映射" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#自同態線性映射"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>自同態線性映射</span> </div> </a> <button aria-controls="toc-自同態線性映射-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>开关自同態線性映射子章节</span> </button> <ul id="toc-自同態線性映射-sublist" class="vector-toc-list"> <li id="toc-自同態和自同構" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#自同態和自同構"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>自同態和自同構</span> </div> </a> <ul id="toc-自同態和自同構-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-自同态在基下矩阵的分类" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#自同态在基下矩阵的分类"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>自同态在基下矩阵的分类</span> </div> </a> <ul id="toc-自同态在基下矩阵的分类-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-核、像和秩-零化度定理" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#核、像和秩-零化度定理"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>核、像和秩-零化度定理</span> </div> </a> <ul id="toc-核、像和秩-零化度定理-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-推廣" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#推廣"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>推廣</span> </div> </a> <ul id="toc-推廣-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-參見" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#參見"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>參見</span> </div> </a> <ul id="toc-參見-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-腳注與參考資料" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#腳注與參考資料"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>腳注與參考資料</span> </div> </a> <button aria-controls="toc-腳注與參考資料-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>开关腳注與參考資料子章节</span> </button> <ul id="toc-腳注與參考資料-sublist" class="vector-toc-list"> <li id="toc-腳注" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#腳注"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>腳注</span> </div> </a> <ul id="toc-腳注-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-腳注所引資料" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#腳注所引資料"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.2</span> <span>腳注所引資料</span> </div> </a> <ul id="toc-腳注所引資料-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-其它參考資料" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#其它參考資料"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.3</span> <span>其它參考資料</span> </div> </a> <ul id="toc-其它參考資料-sublist" class="vector-toc-list"> </ul> </li> </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">线性映射</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。49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-49" 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">49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-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%88%D9%8A%D9%84_%D8%AE%D8%B7%D9%8A" title="تحويل خطي – 阿拉伯语" lang="ar" hreflang="ar" data-title="تحويل خطي" 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-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B5%D0%BD_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D0%BE%D1%80" title="Линеен оператор – 保加利亚语" lang="bg" hreflang="bg" data-title="Линеен оператор" 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-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Aplicaci%C3%B3_lineal" title="Aplicació lineal – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Aplicació lineal" 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/%D9%86%DB%95%D8%AE%D8%B4%DB%95%DB%8C_%DA%BE%DB%8E%DA%B5%DB%8C" title="نەخشەی ھێڵی – 中库尔德语" lang="ckb" hreflang="ckb" data-title="نەخشەی 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/Line%C3%A1rn%C3%AD_zobrazen%C3%AD" title="Lineární zobrazení – 捷克语" lang="cs" hreflang="cs" data-title="Lineární zobrazení" 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%BA%D1%83%C3%A7%D0%B0%D1%80%D1%83" title="Линилле куçару – 楚瓦什语" lang="cv" hreflang="cv" data-title="Линилле куçару" 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-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Line%C3%A6r_funktion" title="Lineær funktion – 丹麦语" lang="da" hreflang="da" data-title="Lineær funktion" data-language-autonym="Dansk" data-language-local-name="丹麦语" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Lineare_Abbildung" title="Lineare Abbildung – 德语" lang="de" hreflang="de" data-title="Lineare Abbildung" 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%93%CF%81%CE%B1%CE%BC%CE%BC%CE%B9%CE%BA%CE%AE_%CE%B1%CF%80%CE%B5%CE%B9%CE%BA%CF%8C%CE%BD%CE%B9%CF%83%CE%B7" title="Γραμμική απεικόνιση – 希腊语" lang="el" hreflang="el" data-title="Γραμμική απεικόνιση" 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/Linear_map" title="Linear map – 英语" lang="en" hreflang="en" data-title="Linear map" 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/Lineara_transformo" title="Lineara transformo – 世界语" lang="eo" hreflang="eo" data-title="Lineara transformo" 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/Aplicaci%C3%B3n_lineal" title="Aplicación lineal – 西班牙语" lang="es" hreflang="es" data-title="Aplicación lineal" 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/Lineaarkujutus" title="Lineaarkujutus – 爱沙尼亚语" lang="et" hreflang="et" data-title="Lineaarkujutus" 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/Aplikazio_lineal" title="Aplikazio lineal – 巴斯克语" lang="eu" hreflang="eu" data-title="Aplikazio lineal" 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/%D9%86%DA%AF%D8%A7%D8%B4%D8%AA_%D8%AE%D8%B7%DB%8C" title="نگاشت خطی – 波斯语" lang="fa" hreflang="fa" data-title="نگاشت 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/Lineaarikuvaus" title="Lineaarikuvaus – 芬兰语" lang="fi" hreflang="fi" data-title="Lineaarikuvaus" 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/Application_lin%C3%A9aire" title="Application linéaire – 法语" lang="fr" hreflang="fr" data-title="Application linéaire" 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/Aplicaci%C3%B3n_linear" title="Aplicación linear – 加利西亚语" lang="gl" hreflang="gl" data-title="Aplicación linear" 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%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="העתקה ליניארית – 希伯来语" lang="he" hreflang="he" data-title="העתקה ליניארית" 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-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Linearni_operator" title="Linearni operator – 克罗地亚语" lang="hr" hreflang="hr" data-title="Linearni operator" 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_lek%C3%A9pez%C3%A9s" title="Lineáris leképezés – 匈牙利语" lang="hu" hreflang="hu" data-title="Lineáris leképezés" 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%A1%D6%80%D5%BF%D5%A1%D5%BA%D5%A1%D5%BF%D5%AF%D5%A5%D6%80%D5%B8%D6%82%D5%B4" title="Գծային արտապատկերում – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Գծային արտապատկերում" 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-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Peta_linear" title="Peta linear – 印度尼西亚语" lang="id" hreflang="id" data-title="Peta 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%ADnuleg_v%C3%B6rpun" title="Línuleg vörpun – 冰岛语" lang="is" hreflang="is" data-title="Línuleg vörpun" 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/Trasformazione_lineare" title="Trasformazione lineare – 意大利语" lang="it" hreflang="it" data-title="Trasformazione lineare" 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%E5%86%99%E5%83%8F" title="線型写像 – 日语" lang="ja" hreflang="ja" data-title="線型写像" 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%84%A0%ED%98%95_%EB%B3%80%ED%99%98" title="선형 변환 – 韩语" lang="ko" hreflang="ko" data-title="선형 변환" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Tiesinis_operatorius" title="Tiesinis operatorius – 立陶宛语" lang="lt" hreflang="lt" data-title="Tiesinis operatorius" data-language-autonym="Lietuvių" data-language-local-name="立陶宛语" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lineaire_afbeelding" title="Lineaire afbeelding – 荷兰语" lang="nl" hreflang="nl" data-title="Lineaire afbeelding" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Line%C3%A6r_transformasjon" title="Lineær transformasjon – 书面挪威语" lang="nb" hreflang="nb" data-title="Lineær transformasjon" 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-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B2%E0%A9%80%E0%A8%A8%E0%A9%80%E0%A8%85%E0%A8%B0_%E0%A8%AE%E0%A9%88%E0%A8%AA" title="ਲੀਨੀਅਰ ਮੈਪ – 旁遮普语" lang="pa" hreflang="pa" data-title="ਲੀਨੀਅਰ ਮੈਪ" 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/Przekszta%C5%82cenie_liniowe" title="Przekształcenie liniowe – 波兰语" lang="pl" hreflang="pl" data-title="Przekształcenie liniowe" 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%A6%D8%B1_%D9%85%DB%8C%D9%BE" title="لینیئر میپ – Western Punjabi" lang="pnb" hreflang="pnb" data-title="لینیئر میپ" 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/Transforma%C3%A7%C3%A3o_linear" title="Transformação linear – 葡萄牙语" lang="pt" hreflang="pt" data-title="Transformação linear" 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/Transformare_liniar%C4%83" title="Transformare liniară – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Transformare liniară" 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%9B%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%BE%D0%B5_%D0%BE%D1%82%D0%BE%D0%B1%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5" title="Линейное отображение – 俄语" lang="ru" hreflang="ru" data-title="Линейное отображение" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Linearna_transformacija" title="Linearna transformacija – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Linearna transformacija" 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-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Linear_mapping" title="Linear mapping – Simple English" lang="en-simple" hreflang="en-simple" data-title="Linear mapping" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Line%C3%A1rne_zobrazenie" title="Lineárne zobrazenie – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Lineárne zobrazenie" 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/Linearna_transformacija" title="Linearna transformacija – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Linearna transformacija" 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-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Harta_lineare" title="Harta lineare – 阿尔巴尼亚语" lang="sq" hreflang="sq" data-title="Harta lineare" data-language-autonym="Shqip" data-language-local-name="阿尔巴尼亚语" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%BE_%D0%BF%D1%80%D0%B5%D1%81%D0%BB%D0%B8%D0%BA%D0%B0%D0%B2%D0%B0%D1%9A%D0%B5" title="Линеарно пресликавање – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Линеарно пресликавање" 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%A4r_avbildning" title="Linjär avbildning – 瑞典语" lang="sv" hreflang="sv" data-title="Linjär avbildning" 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%95%E0%AF%8B%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81" title="நேரியல் கோப்பு – 泰米尔语" lang="ta" hreflang="ta" data-title="நேரியல் கோப்பு" 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-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Do%C4%9Frusal_d%C3%B6n%C3%BC%C5%9F%C3%BCm" title="Doğrusal dönüşüm – 土耳其语" lang="tr" hreflang="tr" data-title="Doğrusal dönüşüm" 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-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9B%D1%96%D0%BD%D1%96%D0%B9%D0%BD%D0%B5_%D0%B2%D1%96%D0%B4%D0%BE%D0%B1%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%BD%D1%8F" title="Лінійне відображення – 乌克兰语" lang="uk" hreflang="uk" data-title="Лінійне відображення" 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/%D9%84%DA%A9%DB%8C%D8%B1%DB%8C_%D8%A7%D8%B3%D8%AA%D8%AD%D8%A7%D9%84%DB%81" title="لکیری استحالہ – 乌尔都语" lang="ur" hreflang="ur" data-title="لکیری استحالہ" 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/Bi%E1%BA%BFn_%C4%91%E1%BB%95i_tuy%E1%BA%BFn_t%C3%ADnh" title="Biến đổi tuyến tính – 越南语" lang="vi" hreflang="vi" data-title="Biến đổi 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-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="线性映射 – 吴语" lang="wuu" hreflang="wuu" data-title="线性映射" data-language-autonym="吴语" data-language-local-name="吴语" class="interlanguage-link-target"><span>吴语</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q207643#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="命名空间"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="浏览条目正文[c]" accesskey="c"><span>条目</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" rel="discussion" title="关于此页面的讨论[t]" accesskey="t"><span>讨论</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown " > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="更改语言变体" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">不转换</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-varlang-0" class="selected ca-variants-zh mw-list-item"><a href="/zh/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" lang="zh" hreflang="zh"><span>不转换</span></a></li><li id="ca-varlang-1" class="ca-variants-zh-Hans mw-list-item"><a href="/zh-hans/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" lang="zh-Hans" hreflang="zh-Hans"><span>简体</span></a></li><li id="ca-varlang-2" class="ca-variants-zh-Hant mw-list-item"><a href="/zh-hant/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" lang="zh-Hant" hreflang="zh-Hant"><span>繁體</span></a></li><li id="ca-varlang-3" class="ca-variants-zh-Hans-CN mw-list-item"><a href="/zh-cn/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" lang="zh-Hans-CN" hreflang="zh-Hans-CN"><span>大陆简体</span></a></li><li id="ca-varlang-4" class="ca-variants-zh-Hant-HK mw-list-item"><a href="/zh-hk/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" lang="zh-Hant-HK" hreflang="zh-Hant-HK"><span>香港繁體</span></a></li><li id="ca-varlang-5" class="ca-variants-zh-Hant-MO mw-list-item"><a href="/zh-mo/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" lang="zh-Hant-MO" hreflang="zh-Hant-MO"><span>澳門繁體</span></a></li><li id="ca-varlang-6" class="ca-variants-zh-Hans-MY mw-list-item"><a href="/zh-my/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" lang="zh-Hans-MY" hreflang="zh-Hans-MY"><span>大马简体</span></a></li><li id="ca-varlang-7" class="ca-variants-zh-Hans-SG mw-list-item"><a href="/zh-sg/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" lang="zh-Hans-SG" hreflang="zh-Hans-SG"><span>新加坡简体</span></a></li><li id="ca-varlang-8" class="ca-variants-zh-Hant-TW mw-list-item"><a href="/zh-tw/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" lang="zh-Hant-TW" hreflang="zh-Hant-TW"><span>臺灣正體</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="查看"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84"><span>阅读</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=history" title="本页面的早前版本。[h]" accesskey="h"><span>查看历史</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="工具" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">工具</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">工具</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">隐藏</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="更多选项" > <div class="vector-menu-heading"> 操作 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84"><span>阅读</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=history"><span>查看历史</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> 常规 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%85%A5%E9%A1%B5%E9%9D%A2/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="列出所有与本页相链的页面[j]" accesskey="j"><span>链入页面</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%87%BA%E6%9B%B4%E6%94%B9/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" rel="nofollow" title="页面链出所有页面的更改[k]" accesskey="k"><span>相关更改</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Project:%E4%B8%8A%E4%BC%A0" title="上传图像或多媒体文件[u]" accesskey="u"><span>上传文件</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:%E7%89%B9%E6%AE%8A%E9%A1%B5%E9%9D%A2" title="全部特殊页面的列表[q]" accesskey="q"><span>特殊页面</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&oldid=84918282" title="此页面该修订版本的固定链接"><span>固定链接</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=info" title="关于此页面的更多信息"><span>页面信息</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:%E5%BC%95%E7%94%A8%E6%AD%A4%E9%A1%B5%E9%9D%A2&page=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&id=84918282&wpFormIdentifier=titleform" title="有关如何引用此页面的信息"><span>引用此页</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:URL%E7%BC%A9%E7%9F%AD%E7%A8%8B%E5%BA%8F&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E7%25BA%25BF%25E6%2580%25A7%25E6%2598%25A0%25E5%25B0%2584"><span>获取短链接</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E7%25BA%25BF%25E6%2580%25A7%25E6%2598%25A0%25E5%25B0%2584"><span>下载二维码</span></a></li> </ul> </div> </div> <div id="p-electronpdfservice-sidebar-portlet-heading" class="vector-menu mw-portlet mw-portlet-electronpdfservice-sidebar-portlet-heading" > <div class="vector-menu-heading"> 打印/导出 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="electron-print_pdf" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=show-download-screen"><span>下载为PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="javascript:print();" rel="alternate" title="本页面的可打印版本[p]" accesskey="p"><span>打印页面</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> 在其他项目中 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Linear_operators" hreflang="en"><span>维基共享资源</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q207643" title="链接到连接的数据仓库项目[g]" accesskey="g"><span>维基数据项目</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="外观"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" 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-e0ef87ba" 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-e0ef87ba" class="noteTA"><div class="noteTA-group"><div data-noteta-group-source="module" data-noteta-group="Math"></div></div><div class="noteTA-local"><div data-noteta-code="zh-cn:數學對象;zh-tw:數學物件;"></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 href="/wiki/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84" title="线性方程组">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 class="mw-selflink selflink">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 class="mw-selflink selflink">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> <p><b>線性映射</b>（英語：<span lang="en">linear map</span>）是<a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%96%93" class="mw-redirect" title="向量空間">向量空間</a>之間，保持向量加法和純量乘法的<a href="/wiki/%E6%98%A0%E5%B0%84" title="映射">函數</a>。線性映射也是向量空間作為模的<a href="/wiki/%E5%90%8C%E6%80%81" title="同态">同態</a><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>。 </p><p><b>線性算子</b>（英語：<span lang="en">linear operator</span>）與<b>線性轉換</b>（英語：<span lang="en">linear transformation</span>，又稱<b>線性變換</b>）是與線性映射相關的慣用名詞，但其實際意義存在許多分歧，詳見<a href="/wiki/%E7%B7%9A%E6%80%A7%E6%98%A0%E5%B0%84#相關名詞" class="mw-redirect" title="線性映射">相關名詞</a>一節。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="正式定義"><span id=".E6.AD.A3.E5.BC.8F.E5.AE.9A.E7.BE.A9"></span>正式定義</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=1" title="编辑章节：正式定義"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>設 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> 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 W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> 都是係數<a href="/wiki/%E5%9F%9F_(%E6%95%B8%E5%AD%B8)" class="mw-redirect" title="域 (數學)">體</a>為 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> 的向量空間， <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/574dffa1c85efaef6b6ef553ebd8ad9cf7f87fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.052ex; height:2.509ex;" alt="{\displaystyle f:V\to W}"></span> 是一個從 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> 送到 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> 的一個<a href="/wiki/%E5%87%BD%E6%95%B8" class="mw-redirect" title="函數">映射</a>。如果 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> 具有以下兩個性質： </p> <ol><li><b>維持向量加法</b>——對於任意兩個 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> 中的向量 <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> 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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> ：<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x+y)=f(x)+f(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x+y)=f(x)+f(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11e072f8427aa606b95bad4d8fba9cb3da2c0b09" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.013ex; height:2.843ex;" alt="{\displaystyle f(x+y)=f(x)+f(y)}"></span></li> <li><b>維持純量乘法</b>——對於任何 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> 中的向量 <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> 和任何純量 <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 a\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f97d838bfcfb39f7a33ffe31cd1c2a989b8ca3f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.136ex; height:2.176ex;" alt="{\displaystyle a\in K}"></span> ：<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a\cdot x)=a\cdot f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(a\cdot x)=a\cdot f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eeea715d963c53b5db41eeadb16f2b1ef61087d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.751ex; height:2.843ex;" alt="{\displaystyle f(a\cdot x)=a\cdot f(x)}"></span></li></ol> <p>則稱 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> 是一個 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span><b>-線性映射</b>。在係數體不致混淆的情況下也經常簡稱<b>線性映射。</b> </p><p>這等價於要求 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> 對任意向量 <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 x_{1},\,\ldots ,\,x_{m}\in V}"> <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" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\,\ldots ,\,x_{m}\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1b4c84c9177df82ca908120d6a8d0ba327a1c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.969ex; height:2.509ex;" alt="{\displaystyle x_{1},\,\ldots ,\,x_{m}\in V}"></span> 和任意純量 <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 a_{1},\,\ldots ,\,a_{m}\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\,\ldots ,\,a_{m}\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81aaf0f8509c446ad914c5519a17eda22a7f5e54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.048ex; height:2.509ex;" alt="{\displaystyle a_{1},\,\ldots ,\,a_{m}\in K}"></span> ：<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a_{1}\cdot x_{1}+\cdots +a_{m}\cdot x_{m})=a_{1}\cdot f(x_{1})+\cdots +a_{m}\cdot f(x_{m})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(a_{1}\cdot x_{1}+\cdots +a_{m}\cdot x_{m})=a_{1}\cdot f(x_{1})+\cdots +a_{m}\cdot f(x_{m})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caeba857aba8776d9366c7d474d9b9952bca46d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.042ex; height:2.843ex;" alt="{\displaystyle f(a_{1}\cdot x_{1}+\cdots +a_{m}\cdot x_{m})=a_{1}\cdot f(x_{1})+\cdots +a_{m}\cdot f(x_{m})}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="線性泛函"><span id=".E7.B7.9A.E6.80.A7.E6.B3.9B.E5.87.BD"></span>線性泛函</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=2" title="编辑章节：線性泛函"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>任何的<a href="/wiki/%E5%9F%9F_(%E6%95%B8%E5%AD%B8)" class="mw-redirect" title="域 (數學)">體</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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> 本身就是一維的（係數為自身的）向量空間，所以可以考慮任何從係數<a href="/wiki/%E5%9F%9F_(%E6%95%B8%E5%AD%B8)" class="mw-redirect" title="域 (數學)">體</a>同樣為 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> 的向量空間 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> 送往 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> 的線性映射，這類線性映射被稱為<b><a href="/wiki/%E7%B7%9A%E6%80%A7%E6%B3%9B%E5%87%BD" title="線性泛函">線性泛函</a></b>。研究線性泛函的學科是線性泛函分析，是<a href="/wiki/%E6%B3%9B%E5%87%BD%E5%88%86%E6%9E%90" title="泛函分析">泛函分析</a>最成熟的分支。 </p> <div class="mw-heading mw-heading3"><h3 id="注意事項"><span id=".E6.B3.A8.E6.84.8F.E4.BA.8B.E9.A0.85"></span>注意事項</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=3" title="编辑章节：注意事項"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>線性映射中的「線性」與「函數圖形是直線」沒有任何關聯。</li></ul> <ul><li>定義域和對應域相同的線性映射可以進行函數合成，合成的結果依然會是線性映射。但是如果改變合成的順序，那合成出來的結果通常不會相同。例如「把函數乘上 <span class="mwe-math-element"><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"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </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/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"></span> 」和「對函數進行微分」都是線性算子，但是對一個函數「先乘上 <span class="mwe-math-element"><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"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </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/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"></span> 再進行微分」和「先進行微分再乘上 <span class="mwe-math-element"><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"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </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/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"></span> 」是不同的線性映射。<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li>維持向量加法的映射可能不維持純量乘法；同樣地，維持純量乘法的映射也可能不維持向量加法。<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="相關名詞"><span id=".E7.9B.B8.E9.97.9C.E5.90.8D.E8.A9.9E"></span>相關名詞</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=4" title="编辑章节：相關名詞"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>線性變換</b>和<b>線性算子</b>這兩個名詞，與本條目的<b>線性映射</b>密切相關，但不同作者有不同的定義。而這種定義分歧的根源在於，如 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:V\rightarrow V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:V\rightarrow V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/082ccfa2dbea1821f4ea50fb3b3315cfed4691cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.404ex; height:2.509ex;" alt="{\displaystyle f:V\rightarrow V}"></span> 這樣，定義域和值域落在同個向量空間的<b>特殊線性映射</b>，有些人為了凸顯而予之不同的稱呼。 </p><p>比如<a href="/wiki/%E8%B0%A2%E5%B0%94%E9%A1%BF%C2%B7%E9%98%BF%E5%85%8B%E6%96%AF%E5%8B%92" title="谢尔顿·阿克斯勒">Axler</a>和<a href="/wiki/%E9%BE%94%E6%98%87" class="mw-redirect" title="龔昇">龔昇</a>就稱這種特殊線性映射為<b>線性算子</b><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Axler_p38_5-0" class="reference"><a href="#cite_note-Axler_p38-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>，但另一方面將<b>線性映射</b>和<b>線性變換</b>視為同義詞；李尚志則將這種特殊線性映射稱為<b>線性變換</b><sup id="cite_ref-李尚志_6-0" class="reference"><a href="#cite_note-李尚志-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>；而<a href="/wiki/%E6%B3%9B%E5%87%BD%E5%88%86%E6%9E%90" title="泛函分析">泛函分析</a>的書籍一般將三者都視為本條目所定義的「<b>線性映射</b>」，其他細節以函數的符號傳達<sup id="cite_ref-柯爾莫哥洛夫_7-0" class="reference"><a href="#cite_note-柯爾莫哥洛夫-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lax_8-0" class="reference"><a href="#cite_note-Lax-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>。 </p><p>本條目採用泛函分析的習慣。 </p> <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%98%A0%E5%B0%84&action=edit&section=5" title="编辑章节：例子"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E6%81%86%E7%AD%89%E5%87%BD%E6%95%B8" title="恆等函數">恒等映射</a>和<a href="/wiki/%E9%9B%B6%E5%87%BD%E6%95%B0" class="mw-redirect" title="零函数">零映射</a>是線性的。<sup id="cite_ref-Axler_page38-39_9-0" class="reference"><a href="#cite_note-Axler_page38-39-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li>對於實數，映射<span class="mwe-math-element"><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\mapsto x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40b49de3850ec5b2be3acb8db45514958c5e80ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.328ex; height:2.676ex;" alt="{\displaystyle x\mapsto x^{2}}"></span>不是線性的。</li></ul> <ul><li>如果<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span></i>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>實<a href="/wiki/%E7%9F%A9%E9%99%A3" class="mw-redirect" title="矩陣">矩陣</a>，則<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span></i>定義了一個從<span class="mwe-math-element"><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^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ebce0f18894435d56a8fe182e22f135aa8ed07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.982ex; height:2.343ex;" alt="{\displaystyle R^{n}}"></span>到<span class="mwe-math-element"><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^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a08a239e7a2966db4af2e766388d97cd839fd831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.439ex; height:2.343ex;" alt="{\displaystyle R^{m}}"></span>的線性映射，這個映射將<a href="/wiki/%E5%88%97%E5%90%91%E9%87%8F" class="mw-redirect" title="列向量">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 x\in R^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in R^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0568155acacb640c51a8d99d214c6b2aee8a4216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.153ex; height:2.343ex;" alt="{\displaystyle x\in R^{n}}"></span>映射到列向量<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 Ax\in R^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>∈</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax\in R^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24f09556c41286d7ebd909f9d92dc84bb1d3bd42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.352ex; height:2.343ex;" alt="{\displaystyle Ax\in R^{m}}"></span>。反過來說，在有限維向量空間之間的任何線性映射都可以用這種方式表示；參見後面章節。</li></ul> <ul><li><a href="/wiki/%E7%A9%8D%E5%88%86" class="mw-redirect" title="積分">積分</a>生成從在某個<a href="/wiki/%E5%8D%80%E9%96%93" title="區間">區間</a>上所有可積分實函數的空間到<span class="mwe-math-element"><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/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>的線性映射。這只是把積分的基本性質（“積分的可加性”和“可從積分號內提出常數倍數”）用另一種說法表述出來。<sup id="cite_ref-Axler_page38-39_9-1" class="reference"><a href="#cite_note-Axler_page38-39-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li><a href="/wiki/%E5%B0%8E%E6%95%B8" class="mw-redirect" title="導數">微分</a>是從所有可微分函數的空間到所有函數的空間的線性映射。<sup id="cite_ref-Axler_page38-39_9-2" class="reference"><a href="#cite_note-Axler_page38-39-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li>“給函數乘上<span class="mwe-math-element"><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"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </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/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"></span>”是一種線性映射。<sup id="cite_ref-Axler_page38-39_9-3" class="reference"><a href="#cite_note-Axler_page38-39-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>設<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>是由全體連續函數所組成的函數空間，則此運算也是空間<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>中的算子。</li></ul> <ul><li>後向移位（backward shift）運算是一種線性映射。即把無窮維向量<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 (x_{1},x_{2},x_{3},x_{4},...)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},x_{2},x_{3},x_{4},...)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7209e6de28c5b8afb71cae1ebec8d606412894d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.583ex; height:2.843ex;" alt="{\displaystyle (x_{1},x_{2},x_{3},x_{4},...)}"></span>的第一個坐標劃去：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {T} (x_{1},x_{2},x_{3},x_{4},...)=(x_{2},x_{3},x_{4},...)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">T</mi> <mo>⁡</mo> <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> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {T} (x_{1},x_{2},x_{3},x_{4},...)=(x_{2},x_{3},x_{4},...)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d36b082c34c86b2a74e0543f59eea2b0584646" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.524ex; height:2.843ex;" alt="{\displaystyle \operatorname {T} (x_{1},x_{2},x_{3},x_{4},...)=(x_{2},x_{3},x_{4},...)}"></span>。<sup id="cite_ref-Axler_page38-39_9-4" class="reference"><a href="#cite_note-Axler_page38-39-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li>如果<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></i>和<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span></i>為在體<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>上的有限維向量空間，則從線性映射<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:V\rightarrow W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:V\rightarrow W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd5c7a685c03396db375d098221baa9b71d76fd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.052ex; height:2.509ex;" alt="{\displaystyle f:V\rightarrow W}"></span>到在後面所描述的<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 \dim _{F}(W)\times \dim _{F}(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>dim</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>W</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msub> <mi>dim</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim _{F}(W)\times \dim _{F}(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5094843f7f3b2b8560885cbed3129ba48af6bf0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.358ex; height:2.843ex;" alt="{\displaystyle \dim _{F}(W)\times \dim _{F}(V)}"></span>矩陣的函數也是線性映射。<sup id="cite_ref-Axler_page38-39_9-5" class="reference"><a href="#cite_note-Axler_page38-39-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li>一次函數<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)=x+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)=x+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ef683918c9f1f6f78d2f316a93c1ddcfd425621" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.938ex; height:2.843ex;" alt="{\displaystyle y=f(x)=x+b}"></span>僅在<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 b=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19206e7d4dab695ccb34c502eff0741e98dbdfc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=0}"></span>時才是一種線性變換。容易驗證一次函數僅在<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 b=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19206e7d4dab695ccb34c502eff0741e98dbdfc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=0}"></span>時，線性變換的基本性質<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(0)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d308c32c9894b88115262081194321ae7d9bbf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(0)=0}"></span>才能成立。（盡管<span class="mwe-math-element"><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\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad073253b4c817f2ec7e3dd7517b7f89a8e581dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.258ex; height:2.676ex;" alt="{\displaystyle b\neq 0}"></span>時其圖像也是一條直線，但<b>這里所說的線性不是指函數圖像為直線</b>。）同理，平移變換一般也不是線性變換（平移距離為零時才是線性變換）。<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="矩陣"><span id=".E7.9F.A9.E9.99.A3"></span>矩陣</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=6" title="编辑章节：矩陣"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>若 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></i> 和 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span></i> 是有限<a href="/wiki/%E5%9F%BA_(%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8)#維度" title="基 (線性代數)">維</a>的、有<b>相同的</b>係數<a href="/wiki/%E5%9F%9F_(%E6%95%B8%E5%AD%B8)" class="mw-redirect" title="域 (數學)">體</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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> 的向量空間，則從 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></i> 到 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span></i> 的線性映射可以用<a href="/wiki/%E7%9F%A9%E9%99%A3" class="mw-redirect" title="矩陣">矩陣</a>表示。 </p> <div class="mw-heading mw-heading3"><h3 id="以矩陣表示線性映射"><span id=".E4.BB.A5.E7.9F.A9.E9.99.A3.E8.A1.A8.E7.A4.BA.E7.B7.9A.E6.80.A7.E6.98.A0.E5.B0.84"></span>以矩陣表示線性映射</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=7" title="编辑章节：以矩陣表示線性映射"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>假設 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c59f606d4e06de82ae3016ab89884480356b3d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.41ex; height:2.176ex;" alt="{\displaystyle T:V\to W}"></span> 是個線性映射，且 </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 {\mathfrak {B}}_{V}=\left\{\alpha _{1},\alpha _{2},\,\ldots ,\alpha _{n}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}_{V}=\left\{\alpha _{1},\alpha _{2},\,\ldots ,\alpha _{n}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e418eccfde9170a40179e242a475a761c9d3a99c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.363ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {B}}_{V}=\left\{\alpha _{1},\alpha _{2},\,\ldots ,\alpha _{n}\right\}}"></span></dd></dl> <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 {\mathfrak {B}}_{W}=\left\{\beta _{1},\beta _{2},\,\ldots ,\beta _{m}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}_{W}=\left\{\beta _{1},\beta _{2},\,\ldots ,\beta _{m}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb4684e60e96524e0708ac4844a0ca978f615bd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.762ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {B}}_{W}=\left\{\beta _{1},\beta _{2},\,\ldots ,\beta _{m}\right\}}"></span></dd></dl> <p>分別是 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></i> 和 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span></i> 的<a href="/wiki/%E5%9F%BA_(%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8)#維度" title="基 (線性代數)">基底</a>。 </p><p>根據基底 <a href="/wiki/%E5%BA%8F%E5%88%97" title="序列"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}_{W}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}_{W}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5e025b84543bac78ed4c5f16f15a5d288d9c65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.009ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {B}}_{W}}"></span></a> 的基本定義，對於每個基向量 <a href="/wiki/%E5%BA%8F%E5%88%97" title="序列"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{i}\in {\mathfrak {B}}_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{i}\in {\mathfrak {B}}_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ca80528650aae470fb4763cbab97d18a705c6e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.678ex; height:2.509ex;" alt="{\displaystyle \alpha _{i}\in {\mathfrak {B}}_{V}}"></span></a> ，存在唯一一組純量 <a href="/wiki/%E5%BA%8F%E5%88%97" title="序列"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1i},\,t_{2i},\,\ldots ,\,t_{mi}\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1i},\,t_{2i},\,\ldots ,\,t_{mi}\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7da0c6bd2e96339fabe9f0674f9cdbe2e595f9aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.285ex; height:2.509ex;" alt="{\displaystyle t_{1i},\,t_{2i},\,\ldots ,\,t_{mi}\in K}"></span></a> 使得 </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 T(\alpha _{i})=\sum _{j=1}^{m}t_{ji}\cdot \beta _{j}=t_{1i}\cdot \beta _{1}+t_{2i}\cdot \beta _{2}+\cdots +t_{mi}\cdot \beta _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\alpha _{i})=\sum _{j=1}^{m}t_{ji}\cdot \beta _{j}=t_{1i}\cdot \beta _{1}+t_{2i}\cdot \beta _{2}+\cdots +t_{mi}\cdot \beta _{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07574a1ca5bcd29d412c73766aa8a5ba446bbced" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:53.91ex; height:7.176ex;" alt="{\displaystyle T(\alpha _{i})=\sum _{j=1}^{m}t_{ji}\cdot \beta _{j}=t_{1i}\cdot \beta _{1}+t_{2i}\cdot \beta _{2}+\cdots +t_{mi}\cdot \beta _{m}}"></span></dd></dl> <p>直觀上，純量 <a href="/wiki/%E5%BA%8F%E5%88%97" title="序列"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1i},\,t_{2i},\,\ldots ,\,t_{mi}\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1i},\,t_{2i},\,\ldots ,\,t_{mi}\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7da0c6bd2e96339fabe9f0674f9cdbe2e595f9aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.285ex; height:2.509ex;" alt="{\displaystyle t_{1i},\,t_{2i},\,\ldots ,\,t_{mi}\in K}"></span></a> 就是對基向量 <a href="/wiki/%E5%BA%8F%E5%88%97" title="序列"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{i}\in {\mathfrak {B}}_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{i}\in {\mathfrak {B}}_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ca80528650aae470fb4763cbab97d18a705c6e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.678ex; height:2.509ex;" alt="{\displaystyle \alpha _{i}\in {\mathfrak {B}}_{V}}"></span></a> 的作用結果 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\alpha _{i})\in W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\alpha _{i})\in W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e96354ca25de2139202c1e403db915c7cf9d140" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.009ex; height:2.843ex;" alt="{\displaystyle T(\alpha _{i})\in W}"></span> ，在基底 <a href="/wiki/%E5%BA%8F%E5%88%97" title="序列"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}_{W}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}_{W}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5e025b84543bac78ed4c5f16f15a5d288d9c65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.009ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {B}}_{W}}"></span></a> 下的諸分量。 </p><p>現在任取一個 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></i> 裡的向量 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.756ex; height:2.176ex;" alt="{\displaystyle v\in V}"></span></i> ，因為基底 <a href="/wiki/%E5%BA%8F%E5%88%97" title="序列"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43eeee2768adcbab448563f6970bb33be08d2bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.55ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {B}}_{V}}"></span></a> 的基本定義，存在唯一一組純量 <a href="/wiki/%E5%BA%8F%E5%88%97" title="序列"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1},\,v_{2},\,\ldots ,\,v_{n}\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1},\,v_{2},\,\ldots ,\,v_{n}\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c798b71f7f485649776f5736277551e38b81b52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.99ex; height:2.509ex;" alt="{\displaystyle v_{1},\,v_{2},\,\ldots ,\,v_{n}\in K}"></span></a> 使得 </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 v=\sum _{i=1}^{n}v_{i}\cdot \alpha _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=\sum _{i=1}^{n}v_{i}\cdot \alpha _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5efcd6ce621dbd0af7f6bead102004005a0a94c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.862ex; height:6.843ex;" alt="{\displaystyle v=\sum _{i=1}^{n}v_{i}\cdot \alpha _{i}}"></span></dd></dl> <p>這樣根據<a href="/wiki/%E6%B1%82%E5%92%8C%E7%AC%A6%E5%8F%B7" title="求和符号">求和符號</a>的性質，可以得到 </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 T(v)=\sum _{i=1}^{n}v_{i}\cdot \left(\sum _{j=1}^{m}t_{ji}\cdot \beta _{j}\right)=\sum _{i=1}^{n}\sum _{j=1}^{m}(t_{ji}v_{i})\cdot \beta _{j}=\sum _{j=1}^{m}\sum _{i=1}^{n}(t_{ji}v_{i})\cdot \beta _{j}=\sum _{j=1}^{m}\left(\sum _{i=1}^{n}t_{ji}v_{i}\right)\cdot \beta _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(v)=\sum _{i=1}^{n}v_{i}\cdot \left(\sum _{j=1}^{m}t_{ji}\cdot \beta _{j}\right)=\sum _{i=1}^{n}\sum _{j=1}^{m}(t_{ji}v_{i})\cdot \beta _{j}=\sum _{j=1}^{m}\sum _{i=1}^{n}(t_{ji}v_{i})\cdot \beta _{j}=\sum _{j=1}^{m}\left(\sum _{i=1}^{n}t_{ji}v_{i}\right)\cdot \beta _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d805608945923b09c719654b827a957467c2925f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:91.384ex; height:7.676ex;" alt="{\displaystyle T(v)=\sum _{i=1}^{n}v_{i}\cdot \left(\sum _{j=1}^{m}t_{ji}\cdot \beta _{j}\right)=\sum _{i=1}^{n}\sum _{j=1}^{m}(t_{ji}v_{i})\cdot \beta _{j}=\sum _{j=1}^{m}\sum _{i=1}^{n}(t_{ji}v_{i})\cdot \beta _{j}=\sum _{j=1}^{m}\left(\sum _{i=1}^{n}t_{ji}v_{i}\right)\cdot \beta _{j}}"></span></dd></dl> <p>然後考慮到 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(v)\in W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(v)\in W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1798423136a4a1a391efe00a2e8d538c581e1fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.849ex; height:2.843ex;" alt="{\displaystyle T(v)\in W}"></span> ，所以根據基底 <a href="/wiki/%E5%BA%8F%E5%88%97" title="序列"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}_{W}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}_{W}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5e025b84543bac78ed4c5f16f15a5d288d9c65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.009ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {B}}_{W}}"></span></a> 的基本定義，存在唯一一組純量 <a href="/wiki/%E5%BA%8F%E5%88%97" title="序列"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1},\,\lambda _{2},\,\ldots ,\,\lambda _{m}\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1},\,\lambda _{2},\,\ldots ,\,\lambda _{m}\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed70a6aa0fdbc0bf63e634fa65c033c16d231aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.129ex; height:2.509ex;" alt="{\displaystyle \lambda _{1},\,\lambda _{2},\,\ldots ,\,\lambda _{m}\in K}"></span></a> 使得 </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 T(v)=\sum _{j=1}^{m}\lambda _{j}\cdot \beta _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(v)=\sum _{j=1}^{m}\lambda _{j}\cdot \beta _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12036283aabce9de5d9487c747b1f42812b4bc98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:17.583ex; height:7.176ex;" alt="{\displaystyle T(v)=\sum _{j=1}^{m}\lambda _{j}\cdot \beta _{j}}"></span></dd></dl> <p>因為這樣的純量 <a href="/wiki/%E5%BA%8F%E5%88%97" title="序列"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1},\,\lambda _{2},\,\ldots ,\,\lambda _{M}\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1},\,\lambda _{2},\,\ldots ,\,\lambda _{M}\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da13380642101b813463b77080cd7f97dca14413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.413ex; height:2.509ex;" alt="{\displaystyle \lambda _{1},\,\lambda _{2},\,\ldots ,\,\lambda _{M}\in K}"></span></a> 是唯一存在的，所以對 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1,\,2,\,\ldots ,\,m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1,\,2,\,\ldots ,\,m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f733866e08b71c3ecde5fc55944aa83914f63c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:15.822ex; height:2.509ex;" alt="{\displaystyle j=1,\,2,\,\ldots ,\,m}"></span> 有 </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 \lambda _{j}=\sum _{i=1}^{n}t_{ji}v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{j}=\sum _{i=1}^{n}t_{ji}v_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ba8ea912895064970088211bfe5e132241d167d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.35ex; height:6.843ex;" alt="{\displaystyle \lambda _{j}=\sum _{i=1}^{n}t_{ji}v_{i}}"></span></dd></dl> <p>考慮到<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>的定義，上式可以改寫為 </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}\lambda _{1}\\\lambda _{2}\\\vdots \\\lambda _{m}\end{bmatrix}}={\begin{bmatrix}t_{11}&t_{12}&\dots &t_{1n}\\t_{21}&t_{22}&\dots &t_{2n}\\\vdots &\vdots &\ddots &\vdots \\t_{m1}&t_{m2}&\dots &t_{mn}\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\\\vdots \\v_{n}\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> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </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> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <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>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\lambda _{1}\\\lambda _{2}\\\vdots \\\lambda _{m}\end{bmatrix}}={\begin{bmatrix}t_{11}&t_{12}&\dots &t_{1n}\\t_{21}&t_{22}&\dots &t_{2n}\\\vdots &\vdots &\ddots &\vdots \\t_{m1}&t_{m2}&\dots &t_{mn}\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\\\vdots \\v_{n}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fade7eaf717a1a8a4f55c8e989db8a48284fb7fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:40.152ex; height:14.176ex;" alt="{\displaystyle {\begin{bmatrix}\lambda _{1}\\\lambda _{2}\\\vdots \\\lambda _{m}\end{bmatrix}}={\begin{bmatrix}t_{11}&t_{12}&\dots &t_{1n}\\t_{21}&t_{22}&\dots &t_{2n}\\\vdots &\vdots &\ddots &\vdots \\t_{m1}&t_{m2}&\dots &t_{mn}\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\\\vdots \\v_{n}\end{bmatrix}}}"></span></dd></dl> <p>也就是說，只要知道 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\alpha _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\alpha _{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf329403872612c05cd340548506b60ec236709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.733ex; height:2.843ex;" alt="{\displaystyle T(\alpha _{i})}"></span> 在 <a href="/wiki/%E5%BA%8F%E5%88%97" title="序列"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}_{W}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}_{W}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5e025b84543bac78ed4c5f16f15a5d288d9c65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.009ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {B}}_{W}}"></span></a> 下的諸分量 <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 t_{ji}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{ji}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7a1d4ed2b16e331cc646030af5f1a5a435bec26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.317ex; height:2.676ex;" alt="{\displaystyle t_{ji}}"></span> ，任意向量 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.756ex; height:2.176ex;" alt="{\displaystyle v\in V}"></span> </i>的作用結果 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08d0491c822c9afd420c0a440dcf4a4cf43ebdf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.573ex; height:2.843ex;" alt="{\displaystyle T(v)}"></span> ，都可以表示為矩陣 <span class="mwe-math-element"><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 {T} ={[t_{ji}]}_{m\times n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} ={[t_{ji}]}_{m\times n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ab47ab1cfe0e625cbac380b72303a3cf5341769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.508ex; height:3.009ex;" alt="{\displaystyle \mathbf {T} ={[t_{ji}]}_{m\times n}}"></span> 與<a href="/wiki/%E8%A1%8C%E5%90%91%E9%87%8F%E8%88%87%E5%88%97%E5%90%91%E9%87%8F" title="行向量與列向量">行向量</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 {v} ={[v_{i}]}_{n\times 1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>×</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ={[v_{i}]}_{n\times 1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c243862439fd601bdac1762e63598ad57ef3135c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.049ex; height:3.009ex;" alt="{\displaystyle \mathbf {v} ={[v_{i}]}_{n\times 1}}"></span> 的乘積。更直觀的來說，矩陣 <span class="mwe-math-element"><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 {T} ={[t_{ji}]}_{m\times n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} ={[t_{ji}]}_{m\times n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ab47ab1cfe0e625cbac380b72303a3cf5341769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.508ex; height:3.009ex;" alt="{\displaystyle \mathbf {T} ={[t_{ji}]}_{m\times n}}"></span> 就是把 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\alpha _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\alpha _{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf329403872612c05cd340548506b60ec236709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.733ex; height:2.843ex;" alt="{\displaystyle T(\alpha _{i})}"></span> 的諸分量沿<b>行</b>（<b>column</b>）擺放所構成的。 </p><p>由上面的推導可以知道，<b>不同的基底</b> <a href="/wiki/%E5%BA%8F%E5%88%97" title="序列"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43eeee2768adcbab448563f6970bb33be08d2bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.55ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {B}}_{V}}"></span></a> 和 <a href="/wiki/%E5%BA%8F%E5%88%97" title="序列"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}_{W}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}_{W}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5e025b84543bac78ed4c5f16f15a5d288d9c65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.009ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {B}}_{W}}"></span></a> 下，矩陣 <span class="mwe-math-element"><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 {T} ={[t_{ji}]}_{m\times n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} ={[t_{ji}]}_{m\times n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ab47ab1cfe0e625cbac380b72303a3cf5341769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.508ex; height:3.009ex;" alt="{\displaystyle \mathbf {T} ={[t_{ji}]}_{m\times n}}"></span> 也<b>不同</b>，為了強調這點，也會將矩陣 <span class="mwe-math-element"><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 {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9593e3b995a1b57c078873a5ea186c7012e1a5ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.859ex; height:2.176ex;" alt="{\displaystyle \mathbf {T} }"></span> 記為 </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 {T} ={[T]}_{{\mathfrak {B}}_{W}}^{{\mathfrak {B}}_{V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} ={[T]}_{{\mathfrak {B}}_{W}}^{{\mathfrak {B}}_{V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4c2e3cbab9350c132edfd924ac5eb747bc39b82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.135ex; height:3.843ex;" alt="{\displaystyle \mathbf {T} ={[T]}_{{\mathfrak {B}}_{W}}^{{\mathfrak {B}}_{V}}}"></span></dd></dl> <p>來強調這種關聯性。 </p><p>若 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:V\to V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:V\to V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0fabb548ced097e03f45cf54dbb066d23d010f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.762ex; height:2.176ex;" alt="{\displaystyle T:V\to V}"></span> ，在同個向量空間 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></i> 通常沒有取不同基底的必要，那上面的推導可以在 <a href="/wiki/%E5%BA%8F%E5%88%97" title="序列"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}_{V}={\mathfrak {B}}_{W}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}_{V}={\mathfrak {B}}_{W}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6555dc224f82d2bf46f16bc5b49f6546b29de435" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.657ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {B}}_{V}={\mathfrak {B}}_{W}}"></span></a> 的前提下進行。這時上式可以進一步簡寫為 </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 {T} ={[T]}_{{\mathfrak {B}}_{V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} ={[T]}_{{\mathfrak {B}}_{V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41b8abb0e9d73aff454eb142eae020602b3aad57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.763ex; height:3.343ex;" alt="{\displaystyle \mathbf {T} ={[T]}_{{\mathfrak {B}}_{V}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="以線性映射表示矩陣"><span id=".E4.BB.A5.E7.B7.9A.E6.80.A7.E6.98.A0.E5.B0.84.E8.A1.A8.E7.A4.BA.E7.9F.A9.E9.99.A3"></span>以線性映射表示矩陣</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=8" title="编辑章节：以線性映射表示矩陣"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>若有由 <span class="mwe-math-element"><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> 個純量構成的矩陣 <span class="mwe-math-element"><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} ={[a_{ij}]}_{m\times n}\in K^{m\times n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={[a_{ij}]}_{m\times n}\in K^{m\times n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d15e42b95288c26514ff078b4e946cff0245ae89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.933ex; height:3.009ex;" alt="{\displaystyle \mathbf {A} ={[a_{ij}]}_{m\times n}\in K^{m\times n}}"></span> ，如果取 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:K^{n\times 1}\to K^{m\times 1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>×</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>×</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:K^{n\times 1}\to K^{m\times 1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaea8db48009c1caae3439edd2f68b444b21669a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.112ex; height:3.009ex;" alt="{\displaystyle f:K^{n\times 1}\to K^{m\times 1}}"></span> 為 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {x} )=\mathbf {A} \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\mathbf {x} )=\mathbf {A} \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0543db67428197d32999f8df7569eb892cb6ba4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.028ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {x} )=\mathbf {A} \mathbf {x} }"></span></dd></dl> <p>其中 </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 {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}}\in K^{n\times 1}}"> <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"> <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> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>×</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}}\in K^{n\times 1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff98fdb5261049c3aed4410afe8d6192823cfb2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:19.163ex; height:13.843ex;" alt="{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}}\in K^{n\times 1}}"></span></dd></dl> <p>因為<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>只有唯一的結果，上面的定義的確符合<a href="/wiki/%E5%87%BD%E6%95%B8" class="mw-redirect" title="函數">函數</a>定義的基本要求。然後考慮 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{n\times 1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>×</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{n\times 1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce32a06ab9c48b8bf31ec6ad0cb6bb3a5920d787" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.413ex; height:2.676ex;" alt="{\displaystyle K^{n\times 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 K^{m\times 1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>×</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{m\times 1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88a42fe07102927ce05b6d0d9984d01a08cfbdca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.869ex; height:2.676ex;" alt="{\displaystyle K^{m\times 1}}"></span> 都可以視為定義在<b>同個</b>純量<a href="/wiki/%E5%9F%9F_(%E6%95%B8%E5%AD%B8)" class="mw-redirect" title="域 (數學)">體</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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> 上的<a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%96%93" class="mw-redirect" title="向量空間">向量空間</a>，而且矩陣乘法是線性的，所以上述定義的函數 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> 的確符合線性映射的基本定義。 </p> <div class="mw-heading mw-heading3"><h3 id="用泛性质做矩阵表示"><span id=".E7.94.A8.E6.B3.9B.E6.80.A7.E8.B4.A8.E5.81.9A.E7.9F.A9.E9.98.B5.E8.A1.A8.E7.A4.BA"></span>用泛性质做矩阵表示</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=9" title="编辑章节：用泛性质做矩阵表示"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>根据<a href="/wiki/%E7%A7%AF_(%E8%8C%83%E7%95%B4%E8%AE%BA)" title="积 (范畴论)">积</a>和<a href="/w/index.php?title=%E4%BD%99%E7%A7%AF_(%E8%8C%83%E7%95%B4%E8%AE%BA)&action=edit&redlink=1" class="new" title="余积 (范畴论)（页面不存在）">余积</a>的泛性质，我们有 </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 \mathrm {Hom} \left(\bigoplus _{i\in I}U_{i},\prod _{j\in J}V_{j}\right)\simeq \prod _{i\in I}\prod _{j\in J}\mathrm {Hom} \left(U_{i},V_{j}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>(</mo> <mrow> <munder> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mi>J</mi> </mrow> </munder> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>≃</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mi>J</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Hom} \left(\bigoplus _{i\in I}U_{i},\prod _{j\in J}V_{j}\right)\simeq \prod _{i\in I}\prod _{j\in J}\mathrm {Hom} \left(U_{i},V_{j}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/845a9c3a3fe57b8653e2ccfe316970a8beec9c0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:45.419ex; height:7.676ex;" alt="{\displaystyle \mathrm {Hom} \left(\bigoplus _{i\in I}U_{i},\prod _{j\in J}V_{j}\right)\simeq \prod _{i\in I}\prod _{j\in J}\mathrm {Hom} \left(U_{i},V_{j}\right).}"></span></dd></dl> <p>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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>-<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>构成的<a href="/wiki/%E8%8C%83%E7%95%B4_(%E6%95%B0%E5%AD%A6)" class="mw-redirect" title="范畴 (数学)">范畴</a>中，有限个线性空间的余积和积是一回事。对于 <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 U,V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>,</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U,V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7681409ec5fffdb272f536757c1211fe0151a9b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.604ex; height:2.509ex;" alt="{\displaystyle U,V}"></span> 的基 <span class="mwe-math-element"><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}}_{1},\dots ,{\mathbf {b}}_{n},{\mathbf {b}}'_{1},\dots ,{\mathbf {b}}'_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {b}}_{1},\dots ,{\mathbf {b}}_{n},{\mathbf {b}}'_{1},\dots ,{\mathbf {b}}'_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab4100002239c1a11dd0d1cae364a57f47f998e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.334ex; height:3.009ex;" alt="{\displaystyle {\mathbf {b}}_{1},\dots ,{\mathbf {b}}_{n},{\mathbf {b}}'_{1},\dots ,{\mathbf {b}}'_{m}}"></span>，取 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}={\mathbf {b}}_{i}K,V_{i}={\mathbf {b}}'_{i}k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>K</mi> <mo>,</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}={\mathbf {b}}_{i}K,V_{i}={\mathbf {b}}'_{i}k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68fbfefa51fe9bb0f11c97e1e4f35ad6b70ea0d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.62ex; height:3.009ex;" alt="{\displaystyle U_{i}={\mathbf {b}}_{i}K,V_{i}={\mathbf {b}}'_{i}k}"></span>，我们有 <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 \mathrm {Hom} (U_{i},V_{j})\simeq K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≃</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Hom} (U_{i},V_{j})\simeq K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fce4242e72daa998d33acc2d27913e671c627ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.501ex; height:3.009ex;" alt="{\displaystyle \mathrm {Hom} (U_{i},V_{j})\simeq K}"></span>，所以左边的线性映射 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Hom} (U,V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Hom} (U,V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd5cb938153d9ea041afc6dbf36db4b7ebb1dd34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.254ex; height:2.843ex;" alt="{\displaystyle \mathrm {Hom} (U,V)}"></span> 就被拆解为了 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nm}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle nm}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2ef477a8b346bb37390e1b888b3ca888d184b61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:1.676ex;" alt="{\displaystyle nm}"></span> 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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> 中的元素，这就是线性映射的矩阵表示。 </p> <div class="mw-heading mw-heading3"><h3 id="用矩陣表示線性映射的原因和好處"><span id=".E7.94.A8.E7.9F.A9.E9.99.A3.E8.A1.A8.E7.A4.BA.E7.B7.9A.E6.80.A7.E6.98.A0.E5.B0.84.E7.9A.84.E5.8E.9F.E5.9B.A0.E5.92.8C.E5.A5.BD.E8.99.95"></span>用矩陣表示線性映射的原因和好處</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=10" title="编辑章节：用矩陣表示線性映射的原因和好處"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li>把線性映射寫成具體而簡明的2維數陣形式後，就成了一種矩陣。進而由線性映射的加法規則和覆合規則來分別定義矩陣的加法規則和乘法規則是很自然的想法。<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>當空間的基變化（坐標系變換）時，線性映射的矩陣也會有規律地變化。在特定的基上研究線性映射，就轉化為對矩陣的研究。利用矩陣的乘法，可以把一些線性系統的方程表達得更緊湊（比如把線性方程組用矩陣表達和研究），也使幾何意義更明顯。矩陣可以<a href="/wiki/%E5%88%86%E5%A1%8A%E7%9F%A9%E9%99%A3" title="分塊矩陣">分塊</a>計算，可以通過適當的變換以“解耦”（把覆雜的變換分解為一些簡單變換的組合）。要求出一個線性變換的<a href="/wiki/%E7%A7%A9_(%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8)" class="mw-redirect" title="秩 (線性代數)">秩</a>，先寫出其矩陣形式幾乎是不可避免的一個步驟。</li> <li>遇到<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x+3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x+3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b053d528f4177e98ac3997c0d41b171deb39ea8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.586ex; height:2.509ex;" alt="{\displaystyle y=x+3}"></span>這樣的加上了1個常量的非線性映射可以通過增加1個維度的方法，把變換映射寫成2×2維的方形矩陣形式，從而在形式上把這一類特殊的非線性映射轉化為線性映射。這個辦法也適用於處理在高維線性變換上多加了一個常向量的情形。這在<a href="/wiki/%E8%A8%88%E7%AE%97%E6%A9%9F%E5%9C%96%E5%BD%A2%E5%AD%B8" class="mw-redirect" title="計算機圖形學">計算機圖形學</a>和剛體理論（及其相關<a href="/wiki/%E6%9C%BA%E6%A2%B0%E5%88%B6%E9%80%A0" title="机械制造">機械制造</a>和<a href="/wiki/%E6%9C%BA%E5%99%A8%E4%BA%BA%E5%AD%A6" title="机器人学">機器人學</a>）中都有大量應用。</li> <li>對角化的矩陣具有諸多優點。線性映射在寫成矩陣後可以進行<a href="/wiki/%E5%AF%B9%E8%A7%92%E5%8C%96" class="mw-redirect" title="对角化">對角化</a>（不能對角化的矩陣可以化簡成接近對角矩陣的<a href="/w/index.php?title=%E6%BA%96%E5%B0%8D%E8%A7%92%E7%9F%A9%E9%99%A3&action=edit&redlink=1" class="new" title="準對角矩陣（页面不存在）">準對角矩陣</a>），從而可以獲得對角化矩陣擁有的獨特優勢（極大地簡化乘法運算，易於分塊，容易看出與基的選取無關的<a href="/wiki/%E4%B8%8D%E8%AE%8A%E9%87%8F" title="不變量">不變量</a>）。比如，對於作用於同一個空間的可對角化的方形矩陣<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span></i>，要求出<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span></i>自乘<span class="mwe-math-element"><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>次後的結果<span class="mwe-math-element"><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^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f40057ac796fcce575670828684300e42bf8a227" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.962ex; height:2.343ex;" alt="{\displaystyle A^{n}}"></span>，一個一個慢慢地乘是很麻煩的事情。而知道對角化技巧的人會發現，在將這矩陣對角化後，其乘法運算會變得格外簡單。實際應用中有很多有意思的問題或解題方法都會涉及到矩陣自乘n次的計算，如1階非齊次線性<a href="/wiki/%E9%80%92%E6%8E%A8%E5%85%B3%E7%B3%BB%E5%BC%8F" class="mw-redirect" title="递推关系式">遞推數列</a><a href="/w/index.php?title=%E9%80%9A%E9%A0%85%E5%85%AC%E5%BC%8F&action=edit&redlink=1" class="new" title="通項公式（页面不存在）">通項公式</a>的線性代數求解法和<a href="/wiki/%E9%A6%AC%E7%88%BE%E5%8F%AF%E5%A4%AB%E9%8F%88" class="mw-redirect" title="馬爾可夫鏈">馬爾可夫鏈</a>的極限狀態（極限分布）的求解。線性代數及矩陣論的一個主要問題就是尋找可使矩陣對角化的條件或者可使矩陣化簡到含很多個0的條件<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>，以便簡化計算（這是主要原因之一）。</li></ol> <div class="mw-heading mw-heading2"><h2 id="線性映射的矩陣的例子"><span id=".E7.B7.9A.E6.80.A7.E6.98.A0.E5.B0.84.E7.9A.84.E7.9F.A9.E9.99.A3.E7.9A.84.E4.BE.8B.E5.AD.90"></span>線性映射的矩陣的例子</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=11" title="编辑章节：線性映射的矩陣的例子"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>二維空間<span class="mwe-math-element"><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^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ce07e278be3e058a6303de8359f8b4a4288264a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.676ex;" alt="{\displaystyle R^{2}}"></span>的線性變換的一些特殊情況有： </p> <ul><li>逆時針<a href="/wiki/%E6%97%8B%E8%BD%89" class="mw-redirect" title="旋轉">旋轉</a>90度： <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}0&-1\\1&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}0&-1\\1&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/472a6a19552569a0059d70cb519e3a057ff5da48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.504ex; height:6.176ex;" alt="{\displaystyle A={\begin{bmatrix}0&-1\\1&0\end{bmatrix}}}"></span></dd></dl></li> <li>逆時針<a href="/wiki/%E6%97%8B%E8%BD%89" class="mw-redirect" title="旋轉">旋轉</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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>度<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>： <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}\cos(\theta )&-\sin(\theta )\\\sin(\theta )&\cos(\theta )\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}\cos(\theta )&-\sin(\theta )\\\sin(\theta )&\cos(\theta )\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3226291b1b690f436ed5d2567ad32305cae308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.332ex; height:6.176ex;" alt="{\displaystyle A={\begin{bmatrix}\cos(\theta )&-\sin(\theta )\\\sin(\theta )&\cos(\theta )\end{bmatrix}}}"></span></dd></dl></li> <li>針對<i>y</i>軸<a href="/wiki/%E5%8F%8D%E5%B0%84_(%E6%95%B0%E5%AD%A6)" title="反射 (数学)">反射</a>： <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20ab12cb4eacac8093a3037ddf6fef598c02b568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.504ex; height:6.176ex;" alt="{\displaystyle A={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}"></span></dd></dl></li> <li>在所有方向上<a href="/wiki/%E7%B8%AE%E6%94%BE" class="mw-redirect" title="縮放">放大</a>2倍： <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}2&0\\0&2\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}2&0\\0&2\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62d0da72acdfc3e2c445c09d698424858d390e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.696ex; height:6.176ex;" alt="{\displaystyle A={\begin{bmatrix}2&0\\0&2\end{bmatrix}}}"></span></dd></dl></li> <li><a href="/wiki/%E9%8C%AF%E5%88%87" class="mw-redirect" title="錯切">水平錯切</a>： <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}1&m\\0&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>m</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}1&m\\0&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb849028d4bad1fc2995c05ddb4a47fbbdfda25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.573ex; height:6.176ex;" alt="{\displaystyle A={\begin{bmatrix}1&m\\0&1\end{bmatrix}}}"></span></dd></dl></li> <li><a href="/w/index.php?title=%E6%93%A0%E5%A3%93%E6%98%A0%E5%B0%84&action=edit&redlink=1" class="new" title="擠壓映射（页面不存在）">擠壓</a>： <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}k&0\\0&1/k\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>k</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}k&0\\0&1/k\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30ba6ee9110cb3031a5b34cf8b8e0e0aa75105c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.118ex; height:6.176ex;" alt="{\displaystyle A={\begin{bmatrix}k&0\\0&1/k\end{bmatrix}}}"></span></dd></dl></li> <li>向<i>y</i>軸<a href="/wiki/%E6%8A%95%E5%BD%B1" class="mw-disambig" title="投影">投影</a>： <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}0&0\\0&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}0&0\\0&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7abab8120211268f4e6edd56ee93d493f9418ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.696ex; height:6.176ex;" alt="{\displaystyle A={\begin{bmatrix}0&0\\0&1\end{bmatrix}}}"></span></dd></dl></li></ul> <div class="mw-heading mw-heading2"><h2 id="從給定線性映射構造新的線性映射"><span id=".E5.BE.9E.E7.B5.A6.E5.AE.9A.E7.B7.9A.E6.80.A7.E6.98.A0.E5.B0.84.E6.A7.8B.E9.80.A0.E6.96.B0.E7.9A.84.E7.B7.9A.E6.80.A7.E6.98.A0.E5.B0.84"></span>從給定線性映射構造新的線性映射</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=12" title="编辑章节：從給定線性映射構造新的線性映射"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>兩個線性映射的<a href="/w/index.php?title=%E8%A6%86%E5%90%88%E6%98%A0%E5%B0%84&action=edit&redlink=1" class="new" title="覆合映射（页面不存在）">覆合映射</a>是線性的：如果<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:V\rightarrow W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:V\rightarrow W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd5c7a685c03396db375d098221baa9b71d76fd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.052ex; height:2.509ex;" alt="{\displaystyle f:V\rightarrow W}"></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 g:W\rightarrow Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>W</mi> <mo stretchy="false">→</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:W\rightarrow Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3703225c4f48f8b29a6ded2cc8d25c060e22a40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.783ex; height:2.509ex;" alt="{\displaystyle g:W\rightarrow Z}"></span>是線性的，則<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f:V\rightarrow Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f:V\rightarrow Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0175df963681fce31dd9d594b6b0d3d1f2a1d339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.608ex; height:2.509ex;" alt="{\displaystyle g\circ f:V\rightarrow Z}"></span>也是線性的。 </p><p>若線性映射<a href="/wiki/%E5%8F%8D%E5%87%BD%E6%95%B8" title="反函數">可逆</a>，則該線性映射的<a href="/wiki/%E5%8F%8D%E5%87%BD%E6%95%B8" title="反函數">逆</a>也是線性映射。 </p><p>如果<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}:V\rightarrow W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}:V\rightarrow W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d58ce0893d9490d9311564cdc40a216b96c07f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.967ex; height:2.509ex;" alt="{\displaystyle f_{1}:V\rightarrow W}"></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 f_{2}:V\rightarrow W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{2}:V\rightarrow W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9829d9f0d1740141ea0f19151903203376a68add" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.967ex; height:2.509ex;" alt="{\displaystyle f_{2}:V\rightarrow W}"></span>是線性的，則它們的和<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 f_{1}+f_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}+f_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7875d43d94bfdb30c7a2a5d72fea4a6960bc8787" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.227ex; height:2.509ex;" alt="{\displaystyle f_{1}+f_{2}}"></span>也是線性的(這是由<span class="mwe-math-element"><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(f_{1}+f_{2}\right)\left(x\right)=f_{1}\left(x\right)+f_{2}\left(x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(f_{1}+f_{2}\right)\left(x\right)=f_{1}\left(x\right)+f_{2}\left(x\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dd80e747af0cfbbeca929fa698a909467059c53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.941ex; height:2.843ex;" alt="{\displaystyle \left(f_{1}+f_{2}\right)\left(x\right)=f_{1}\left(x\right)+f_{2}\left(x\right)}"></span>定義的)。 </p><p>如果<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:V\rightarrow W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:V\rightarrow W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd5c7a685c03396db375d098221baa9b71d76fd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.052ex; height:2.509ex;" alt="{\displaystyle f:V\rightarrow W}"></span>是線性的，而<i>a</i>是基礎體<i>K</i>的一個元素，則定義自 (<i>af</i>)(<i>x</i>) = <i>a</i> (<i>f</i>(<i>x</i>))的映射<i>af</i>也是線性的。 </p><p>所以從<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>到<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>的線性映射的集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\left(V,W\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo>,</mo> <mi>W</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\left(V,W\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c963013f69b3284606f130e32962ccd65edeba9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.036ex; height:2.843ex;" alt="{\displaystyle L\left(V,W\right)}"></span>自身形成在<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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>上的向量空間，有時指示為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Hom} \left(V,W\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo>,</mo> <mi>W</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Hom} \left(V,W\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7a68b15446aa7e2f2d5836f4fd9359f19f24aa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.294ex; height:2.843ex;" alt="{\displaystyle \mathrm {Hom} \left(V,W\right)}"></span>。進一步的說，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 V=W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/740038d36bd79466d6938d73b83fe737161fa1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.321ex; height:2.176ex;" alt="{\displaystyle V=W}"></span>的情況中，這個向量空間(指示為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {End} (V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {End} (V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7708ca3ed32f63af36fb62f848320792dace0cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.764ex; height:2.843ex;" alt="{\displaystyle \mathrm {End} (V)}"></span>)是在<a href="/w/index.php?title=%E8%A6%86%E5%90%88%E5%87%BD%E6%95%B8&action=edit&redlink=1" class="new" title="覆合函數（页面不存在）">映射覆合</a>下的<a href="/wiki/%E7%B5%90%E5%90%88%E4%BB%A3%E6%95%B8" title="結合代數">結合代數</a>，因為兩個線性映射的覆合再次是線性映射，所以映射的覆合總是結合律的。 </p><p>給定有限維的情況，如果基已經選擇好了，則線性映射的覆合對應於<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>，線性映射的加法對應於<a href="/wiki/%E7%9F%A9%E9%99%A3%E5%8A%A0%E6%B3%95" title="矩陣加法">矩陣加法</a>，而線性映射與純量的乘法對應於矩陣與純量的乘法。 </p> <div class="mw-heading mw-heading2"><h2 id="自同態線性映射"><span id=".E8.87.AA.E5.90.8C.E6.85.8B.E7.B7.9A.E6.80.A7.E6.98.A0.E5.B0.84"></span>自同態線性映射</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=13" title="编辑章节：自同態線性映射"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r83732972">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}html.client-js body.skin-minerva .mw-parser-output .mbox-text-span{margin-left:23px!important}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .ambox{border-left-color:#36c!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-speedy,html.skin-theme-clientpref-night .mw-parser-output .ambox-delete{border-left-color:#b32424!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-speedy{background-color:#300!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-content{border-left-color:#f28500!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-style{border-left-color:#fc3!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-move{border-left-color:#9932cc!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-protection{border-left-color:#a2a9b1!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .ambox{border-left-color:#36c!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy,html.skin-theme-clientpref-os .mw-parser-output .ambox-delete{border-left-color:#b32424!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy{background-color:#300!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-content{border-left-color:#f28500!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-style{border-left-color:#fc3!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-move{border-left-color:#9932cc!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-protection{border-left-color:#a2a9b1!important}}</style><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">此章节需要<a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit"><b>扩充</b></a>。 <small class="date-container"><i>(<span class="date">2016年6月2日</span>)</i></small></div></td></tr></tbody></table> <p>自同態的線性映射在泛函分析和<a href="/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8" class="mw-redirect" title="量子力學">量子力學</a>中都有很重要的地位。按前文約定，我們用“線性算子”來簡稱它。（注意泛函分析中所說的“線性算子”不一定是<a href="/wiki/%E8%87%AA%E5%90%8C%E6%80%81" title="自同态">自同態</a>(endomorphism)映射，但我們為了照顧不同書籍的差異以及敘述的方便，暫用“線性算子”來稱呼這種自同態。） </p> <div class="mw-heading mw-heading3"><h3 id="自同態和自同構"><span id=".E8.87.AA.E5.90.8C.E6.85.8B.E5.92.8C.E8.87.AA.E5.90.8C.E6.A7.8B"></span>自同態和自同構</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=14" title="编辑章节：自同態和自同構"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E8%87%AA%E5%90%8C%E6%80%81" title="自同态">自同態</a>是一個<a href="/wiki/%E6%95%B0%E5%AD%A6%E5%AF%B9%E8%B1%A1" title="数学对象">數學對象</a>到它本身的保持結構的映射（<a href="/wiki/%E5%90%8C%E6%80%81" title="同态">同態</a>），例如群<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>的自同態則是<a href="/wiki/%E7%BE%A4%E5%90%8C%E6%85%8B" title="群同態">群同態</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 f:G\to G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:G\to G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17f1d5bde362fa99070774ba0f19a6bb93000ec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.483ex; height:2.509ex;" alt="{\displaystyle f:G\to G}"></span>。對於向量空間<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>，其自同態是線性算子<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:V\rightarrow V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:V\rightarrow V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/082ccfa2dbea1821f4ea50fb3b3315cfed4691cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.404ex; height:2.509ex;" alt="{\displaystyle f:V\rightarrow V}"></span>；所有這種自同態的集合<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {End} (V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {End} (V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7708ca3ed32f63af36fb62f848320792dace0cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.764ex; height:2.843ex;" alt="{\displaystyle \mathrm {End} (V)}"></span>與如上定義的加法、覆合和純量乘法一起形成一個<a href="/wiki/%E7%B5%90%E5%90%88%E4%BB%A3%E6%95%B8" title="結合代數">結合代數</a>，帶有在體<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>上的單位元(特別是一個環)。這個代數的乘法單位元是<a href="/wiki/%E6%81%92%E7%AD%89%E6%98%A0%E5%B0%84" class="mw-redirect" title="恒等映射">恒等映射</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 \mathrm {id} :V\rightarrow V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {id} :V\rightarrow V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a1fb2f5c03f74c6b6f53ed6b5aa5ac6a8950515" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.065ex; height:2.176ex;" alt="{\displaystyle \mathrm {id} :V\rightarrow V}"></span>。 </p><p>若<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></i>的自同態也剛好是<a href="/wiki/%E5%90%8C%E6%A7%8B" class="mw-redirect" title="同構">同構</a>則稱之為<a href="/wiki/%E8%87%AA%E5%90%8C%E6%9E%84" title="自同构">自同構</a>。兩個自同構的<a href="/wiki/%E5%A4%8D%E5%90%88%E5%87%BD%E6%95%B0" title="复合函数">覆合</a>再次是自同構，所以<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></i>的所有的自同構的集合形成一個<a href="/wiki/%E7%BE%A4" title="群">群</a>，<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></i>的<a href="/wiki/%E8%87%AA%E5%90%8C%E6%A7%8B%E7%BE%A4" class="mw-redirect" title="自同構群">自同構群</a>可表為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Aut} (V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">t</mi> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Aut} (V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c3f5e3555df7dcfdc0c3a37e3b9e2c97b6e5d54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.537ex; height:2.843ex;" alt="{\displaystyle \mathrm {Aut} (V)}"></span>或<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GL} (V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">L</mi> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {GL} (V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e9f3c8d0e7c42fb9ef98af46cf0c49f81eb1c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.874ex; height:2.843ex;" alt="{\displaystyle \mathrm {GL} (V)}"></span>。因為自同構正好是那些在覆合運算下擁有逆元的自同態，所以<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Aut} (V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">t</mi> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Aut} (V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c3f5e3555df7dcfdc0c3a37e3b9e2c97b6e5d54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.537ex; height:2.843ex;" alt="{\displaystyle \mathrm {Aut} (V)}"></span>也就是在環<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {End} (V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {End} (V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7708ca3ed32f63af36fb62f848320792dace0cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.764ex; height:2.843ex;" alt="{\displaystyle \mathrm {End} (V)}"></span>中的<a href="/wiki/%E5%8F%AF%E9%80%86%E5%85%83" title="可逆元">可逆元群</a>。 </p><p>如果<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></i>之維度<span class="mwe-math-element"><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>有限<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {End} (V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {End} (V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7708ca3ed32f63af36fb62f848320792dace0cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.764ex; height:2.843ex;" alt="{\displaystyle \mathrm {End} (V)}"></span><a href="/wiki/%E5%90%8C%E6%A7%8B" class="mw-redirect" title="同構">同構</a>於帶有在<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span></i>中元素的所有<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\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>矩陣構成的<a href="/wiki/%E7%B5%90%E5%90%88%E4%BB%A3%E6%95%B8" title="結合代數">結合代數</a>，且<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></i>的自同態群<a href="/wiki/%E7%BE%A4%E5%90%8C%E6%A7%8B" title="群同構">同構</a>於帶有在<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span></i>中元素的所有<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\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>可逆矩陣構成的<a href="/wiki/%E4%B8%80%E8%88%AC%E7%B7%9A%E6%80%A7%E7%BE%A4" class="mw-redirect" title="一般線性群">一般線性群</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 \mathrm {GL} (n,K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">L</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {GL} (n,K)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19dd1d99f5551a5d11bde5049faf4a7fa4cbad7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.581ex; height:2.843ex;" alt="{\displaystyle \mathrm {GL} (n,K)}"></span>。 </p> <div class="mw-heading mw-heading3"><h3 id="自同态在基下矩阵的分类"><span id=".E8.87.AA.E5.90.8C.E6.80.81.E5.9C.A8.E5.9F.BA.E4.B8.8B.E7.9F.A9.E9.98.B5.E7.9A.84.E5.88.86.E7.B1.BB"></span>自同态在基下矩阵的分类</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=15" title="编辑章节：自同态在基下矩阵的分类"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E8%8B%A5%E5%B0%94%E5%BD%93%E6%A0%87%E5%87%86%E5%9E%8B" title="若尔当标准型">若尔当标准型</a>叙述了代数闭域 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> 上的线性空间 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> 上的自同态 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> 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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> 的基上的矩阵表示的表现，<a href="/w/index.php?title=%E6%9C%89%E7%90%86%E6%A0%87%E5%87%86%E5%9E%8B&action=edit&redlink=1" class="new" title="有理标准型（页面不存在）">有理标准型</a>是将其推广到任意域上的方法。 </p> <div class="mw-heading mw-heading2"><h2 id="核、像和秩-零化度定理"><span id=".E6.A0.B8.E3.80.81.E5.83.8F.E5.92.8C.E7.A7.A9-.E9.9B.B6.E5.8C.96.E5.BA.A6.E5.AE.9A.E7.90.86"></span>核、像和秩-零化度定理</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=16" title="编辑章节：核、像和秩-零化度定理"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>對於一個線性映射 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:V\rightarrow W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:V\rightarrow W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd5c7a685c03396db375d098221baa9b71d76fd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.052ex; height:2.509ex;" alt="{\displaystyle f:V\rightarrow W}"></span> ，可以考慮以下兩個： </p> <ul><li><b>核</b>（ Kernel ）——送到零向量的那些向量：<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Ker} (f):=f^{-1}(\mathbf {0} )=\{x\in V\mid f(x)=\mathbf {0} \}\subseteq V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>V</mi> <mo>∣</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>⊆</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Ker} (f):=f^{-1}(\mathbf {0} )=\{x\in V\mid f(x)=\mathbf {0} \}\subseteq V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb28dd456b8c76c08c69398d86c0ab7ce4b1e19a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.441ex; height:3.176ex;" alt="{\displaystyle \mathrm {Ker} (f):=f^{-1}(\mathbf {0} )=\{x\in V\mid f(x)=\mathbf {0} \}\subseteq V}"></span></li> <li><b>像</b>（ Image ）——把整個空間送過去後的結果：<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Im} (f):=f(V)=\{f(x)\in W\mid x\in V\}\subseteq W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">m</mi> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>W</mi> <mo>∣</mo> <mi>x</mi> <mo>∈</mo> <mi>V</mi> <mo fence="false" stretchy="false">}</mo> <mo>⊆</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Im} (f):=f(V)=\{f(x)\in W\mid x\in V\}\subseteq W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9faca6299664f8f87a91848e8e45793c4fa5cb1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.029ex; height:2.843ex;" alt="{\displaystyle \mathrm {Im} (f):=f(V)=\{f(x)\in W\mid x\in V\}\subseteq W}"></span></li></ul> <p>那麼 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ker} (f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Ker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Ker} (f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d0cf47c65365246bf08b2441afb3357f0aeaabe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.84ex; height:2.843ex;" alt="{\displaystyle \operatorname {Ker} (f)}"></span> 是 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></i> 的<a href="/wiki/%E7%B7%9A%E6%80%A7%E5%AD%90%E7%A9%BA%E9%96%93" class="mw-redirect" title="線性子空間">子空間</a>，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 \operatorname {Im} (f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} (f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6507f25276d69074273e1fd9039ef6b0ec3f7da9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.863ex; height:2.843ex;" alt="{\displaystyle \operatorname {Im} (f)}"></span> 是 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span></i> 的子空間。下面的叫做<a href="/wiki/%E7%A7%A9-%E9%9B%B6%E5%8C%96%E5%BA%A6%E5%AE%9A%E7%90%86" class="mw-redirect" title="秩-零化度定理">秩-零化度定理</a>的維度公式經常是有用的： </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim(\mathrm {Ker} (f))+\dim(\mathrm {Im} (f))=\dim(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">m</mi> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(\mathrm {Ker} (f))+\dim(\mathrm {Im} (f))=\dim(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04f8e284fecfaa1b4226b31c5038f7011a7765f1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.483ex; height:2.843ex;" alt="{\displaystyle \dim(\mathrm {Ker} (f))+\dim(\mathrm {Im} (f))=\dim(V)}"></span> </p><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 \dim(\mathrm {Im} (f))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">m</mi> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(\mathrm {Im} (f))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67940543b7ff3ce4ae674826739ccf2ac038e6ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.548ex; height:2.843ex;" alt="{\displaystyle \dim(\mathrm {Im} (f))}"></span>這個數稱做「 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> 的秩」（ rank ）並寫成 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {rk} (f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">k</mi> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {rk} (f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e999720b285aa574397333dbe3056675513ec5c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.227ex; height:2.843ex;" alt="{\displaystyle \mathrm {rk} (f)}"></span> ，有時也寫成 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99154d2acec1abd8430087e760b5194734f868dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.29ex; height:2.843ex;" alt="{\displaystyle \rho (f)}"></span> ；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 \dim(\mathrm {Ker} (f))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(\mathrm {Ker} (f))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58fbd35be66405863dd0f96b88fd5d7a6d255045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.525ex; height:2.843ex;" alt="{\displaystyle \dim(\mathrm {Ker} (f))}"></span> 這個數則稱做「 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> 的零化度」（ nullity ）並寫成 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/459dbb6a50c2434c042712d02a089de0d028c0cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.215ex; height:2.843ex;" alt="{\displaystyle v(f)}"></span> 。如果 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></i> 和 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span></i> 是有限維的，那麼 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span></i> 的秩和零化度就是 <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span></i> 的矩陣形式的<a href="/wiki/%E7%9F%A9%E9%99%A3%E7%9A%84%E7%A7%A9" class="mw-redirect" title="矩陣的秩">秩</a>和<a href="/wiki/%E9%9B%B6%E7%A9%BA%E9%96%93#性質" class="mw-redirect" title="零空間">零化度</a>。 </p><p>這個定理在抽象代數的推廣是<a href="/wiki/%E5%90%8C%E6%9E%84%E5%9F%BA%E6%9C%AC%E5%AE%9A%E7%90%86" title="同构基本定理">同構定理</a>。 </p> <div class="mw-heading mw-heading2"><h2 id="推廣"><span id=".E6.8E.A8.E5.BB.A3"></span>推廣</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=17" title="编辑章节：推廣"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E5%A4%9A%E9%87%8D%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="多重线性映射">多重線性映射</a>是線性映射最重要的推廣，它也是<a href="/wiki/%E6%A0%BC%E6%8B%89%E6%96%AF%E6%9B%BC%E4%BB%A3%E6%95%B0" class="mw-redirect" title="格拉斯曼代数">格拉斯曼代數</a>和<a href="/wiki/%E5%BC%A0%E9%87%8F%E5%88%86%E6%9E%90" class="mw-redirect" title="张量分析">張量分析</a>的數學基礎。其特例為<a href="/wiki/%E5%8F%8C%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="双线性映射">雙線性映射</a>。 </p> <div class="mw-heading mw-heading2"><h2 id="參見"><span id=".E5.8F.83.E8.A6.8B"></span>參見</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=18" title="编辑章节：參見"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E7%B7%9A%E6%80%A7%E6%96%B9%E7%A8%8B" class="mw-redirect" title="線性方程">線性方程</a></li> <li><a href="/wiki/%E5%8F%8D%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="反线性映射">反線性映射</a></li> <li><a href="/wiki/%E5%8F%98%E6%8D%A2%E7%9F%A9%E9%98%B5" title="变换矩阵">變換矩陣</a></li> <li><a href="/wiki/%E8%BF%9E%E7%BB%AD%E7%BA%BF%E6%80%A7%E7%AE%97%E5%AD%90" title="连续线性算子">連續線性算子</a></li> <li><a href="/wiki/%E4%BA%BA%E5%B7%A5%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C" title="人工神经网络">人工神經網路</a></li> <li><a href="/wiki/%E8%A8%88%E7%AE%97%E6%A9%9F%E5%9C%96%E5%BD%A2%E5%AD%B8" class="mw-redirect" title="計算機圖形學">計算機圖形學</a></li> <li><a href="/wiki/%E7%B7%9A%E6%80%A7%E7%B3%BB%E7%B5%B1" title="線性系統">線性系統</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="腳注與參考資料"><span id=".E8.85.B3.E6.B3.A8.E8.88.87.E5.8F.83.E8.80.83.E8.B3.87.E6.96.99"></span>腳注與參考資料</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=19" title="编辑章节：腳注與參考資料"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="腳注"><span id=".E8.85.B3.E6.B3.A8"></span>腳注</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=20" title="编辑章节：腳注"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist columns references-column-count references-column-count-2" style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2; list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">見<a href="#CITEREFLax2010">Lax 2010</a>，第7頁(位於第2章“線性映射”第1節“線性映射生成的代數”)。</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">見<a href="#CITEREFAxler2009">Axler 2009</a>，第41頁(位於第3章“線性映射”第1節“定義與例子”)。</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">見<a href="#CITEREFAxler2009">Axler 2009</a>，第59頁(位於第3章“線性映射”末尾習題旁的說明)。</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">見龔昇《線性代數五講》第1講第10頁。</span> </li> <li id="cite_note-Axler_p38-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Axler_p38_5-0">^</a></b></span> <span class="reference-text">見<a href="#CITEREFAxler2009">Axler 2009</a>，第38頁(位於第3章“線性映射”第1節“定義與例子”)。</span> </li> <li id="cite_note-李尚志-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-李尚志_6-0">^</a></b></span> <span class="reference-text"><cite class="citation book">李尚志. 第6章“線性變換”第4節“線性變換”. 線性代數第1版. <a href="/wiki/%E9%AB%98%E7%AD%89%E6%95%99%E8%82%B2%E5%87%BA%E7%89%88%E7%A4%BE" title="高等教育出版社">高等教育出版社</a>. 2006: 326. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/7-04-019870-3" title="Special:网络书源/7-04-019870-3"><span title="国际标准书号">ISBN</span> 7-04-019870-3</a>. <q>則V到自身的線性映射稱為V的線性變換(linear transformation)。</q></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&rft.atitle=%E7%AC%AC6%E7%AB%A0%E2%80%9C%E7%B7%9A%E6%80%A7%E8%AE%8A%E6%8F%9B%E2%80%9D%E7%AC%AC4%E7%AF%80%E2%80%9C%E7%B7%9A%E6%80%A7%E8%AE%8A%E6%8F%9B%E2%80%9D&rft.au=%E6%9D%8E%E5%B0%9A%E5%BF%97&rft.btitle=%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8&rft.date=2006&rft.edition=%E7%AC%AC1%E7%89%88&rft.genre=bookitem&rft.isbn=7-04-019870-3&rft.pages=326&rft.pub=%E9%AB%98%E7%AD%89%E6%95%99%E8%82%B2%E5%87%BA%E7%89%88%E7%A4%BE&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-柯爾莫哥洛夫-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-柯爾莫哥洛夫_7-0">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/%E5%AE%89%E5%BE%B7%E9%9B%B7%C2%B7%E6%9F%AF%E7%88%BE%E8%8E%AB%E5%93%A5%E6%B4%9B%E5%A4%AB" class="mw-redirect" title="安德雷·柯爾莫哥洛夫">А·Н·柯爾莫哥洛夫</a>，佛明(С. В. Фомин). 第4章“線性泛函與線性算子”第5節“線性算子”. Элементы теории функций и функционального анализа [函數論與泛函分析初步]. 俄羅斯數學教材選譯. 段虞榮 (翻譯)，鄭洪深 (翻譯)，郭思旭 (翻譯) 原書第7版，中譯本第2版. 高等教育出版社. 2006年: 162. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/7-04-018407-9" title="Special:网络书源/7-04-018407-9"><span title="国际标准书号">ISBN</span> 7-04-018407-9</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&rft.atitle=%E7%AC%AC4%E7%AB%A0%E2%80%9C%E7%B7%9A%E6%80%A7%E6%B3%9B%E5%87%BD%E8%88%87%E7%B7%9A%E6%80%A7%E7%AE%97%E5%AD%90%E2%80%9D%E7%AC%AC5%E7%AF%80%E2%80%9C%E7%B7%9A%E6%80%A7%E7%AE%97%E5%AD%90%E2%80%9D&rft.au=%D0%90%C2%B7%D0%9D%C2%B7%E6%9F%AF%E7%88%BE%E8%8E%AB%E5%93%A5%E6%B4%9B%E5%A4%AB%EF%BC%8C%E4%BD%9B%E6%98%8E%28%D0%A1.+%D0%92.+%D0%A4%D0%BE%D0%BC%D0%B8%D0%BD%29&rft.btitle=%D0%AD%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D1%8B+%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D0%B8+%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B9+%D0%B8+%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B3%D0%BE+%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0&rft.date=2006&rft.edition=%E5%8E%9F%E6%9B%B8%E7%AC%AC7%E7%89%88%EF%BC%8C%E4%B8%AD%E8%AD%AF%E6%9C%AC%E7%AC%AC2%E7%89%88&rft.genre=bookitem&rft.isbn=7-04-018407-9&rft.pages=162&rft.pub=%E9%AB%98%E7%AD%89%E6%95%99%E8%82%B2%E5%87%BA%E7%89%88%E7%A4%BE&rft.series=%E4%BF%84%E7%BE%85%E6%96%AF%E6%95%B8%E5%AD%B8%E6%95%99%E6%9D%90%E9%81%B8%E8%AD%AF&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Lax-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lax_8-0">^</a></b></span> <span class="reference-text">見<a href="#CITEREFLax2010">Lax 2010</a>，第131頁(位於第15章“有界線性映射”的開頭部分)。原文為“線性映射也稱為線性算子或線性變換”。</span> </li> <li id="cite_note-Axler_page38-39-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-Axler_page38-39_9-0"><sup><b>9.0</b></sup></a> <a href="#cite_ref-Axler_page38-39_9-1"><sup><b>9.1</b></sup></a> <a href="#cite_ref-Axler_page38-39_9-2"><sup><b>9.2</b></sup></a> <a href="#cite_ref-Axler_page38-39_9-3"><sup><b>9.3</b></sup></a> <a href="#cite_ref-Axler_page38-39_9-4"><sup><b>9.4</b></sup></a> <a href="#cite_ref-Axler_page38-39_9-5"><sup><b>9.5</b></sup></a></span> <span class="reference-text">見<a href="#CITEREFAxler2009">Axler 2009</a>，第38-39頁(位於第3章“線性映射”第1節“定義與例子”)。</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">見<a href="#CITEREFArtin2010">Artin 2010</a>，第156頁。(位於第6章“Symmetry”第1節“ Symmetry of the Plane Figures”)</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/%E6%B2%83%E7%88%BE%E7%89%B9%C2%B7%E9%AD%AF%E4%B8%81" title="沃爾特·魯丁">Walter Rudin</a>. 第1章“Topological Vector Spaces”中的“Linear mappings”一節. <a rel="nofollow" class="external text" href="https://archive.org/details/functionalanalys00rudi_008">Functional Analysis [泛函分析]</a>. Higher mathematics series. <a href="/wiki/%E9%BA%A6%E6%A0%BC%E5%8A%B3-%E5%B8%8C%E5%B0%94%E9%9B%86%E5%9B%A2" class="mw-redirect" title="麦格劳-希尔集团">McGraw-Hill Book Company</a>. 1973: <a rel="nofollow" class="external text" href="https://archive.org/details/functionalanalys00rudi_008/page/n22">13</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&rft.atitle=%E7%AC%AC1%E7%AB%A0%E2%80%9CTopological+Vector+Spaces%E2%80%9D%E4%B8%AD%E7%9A%84%E2%80%9CLinear+mappings%E2%80%9D%E4%B8%80%E7%AF%80&rft.au=Walter+Rudin&rft.btitle=Functional+Analysis&rft.date=1973&rft.genre=bookitem&rft.pages=13&rft.pub=McGraw-Hill+Book+Company&rft.series=Higher+mathematics+series&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffunctionalanalys00rudi_008&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">見<a href="#CITEREFAxler2009">Axler 2009</a>，第51頁(位於第3章“線性映射”第3節“線性映射的矩陣”)。</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">見<a href="#CITEREFAxler2009">Axler 2009</a>，第82頁(位於第5章“本征值與本征向量”第3節“上三角矩陣”)。</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">其證明只需要用到<a href="/wiki/%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B8" class="mw-redirect" title="三角函數">三角函數</a>的基礎知識，在網上很容易找到證明過程。也可參見<a href="#CITEREFFeynman">Feynman</a>第11章“Vectors”第3節“Rotations”。</span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="腳注所引資料"><span id=".E8.85.B3.E6.B3.A8.E6.89.80.E5.BC.95.E8.B3.87.E6.96.99"></span>腳注所引資料</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=21" title="编辑章节：腳注所引資料"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFArtin2010" class="citation book"><a href="/wiki/%E8%BF%88%E5%85%8B%E5%B0%94%C2%B7%E9%98%BF%E5%BB%B7" title="迈克尔·阿廷">Michael Artin</a>. <span></span><i>Algebra</i><span></span> [代數] 2. <a href="/wiki/%E5%9F%B9%E7%94%9F%E6%95%99%E8%82%B2" title="培生教育">Pearson</a>. 2010. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0132413770" title="Special:网络书源/978-0132413770"><span title="国际标准书号">ISBN</span> 978-0132413770</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&rft.au=Michael+Artin&rft.btitle=Algebra&rft.date=2010&rft.edition=2&rft.genre=book&rft.isbn=978-0132413770&rft.pub=Pearson&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFAxler2009" class="citation book"><a href="/wiki/%E8%B0%A2%E5%B0%94%E9%A1%BF%C2%B7%E9%98%BF%E5%85%8B%E6%96%AF%E5%8B%92" title="谢尔顿·阿克斯勒">Sheldon Axler</a>. <span></span><i>Linear Algebra Done Right</i><span></span> [線性代數應該這樣學]. 圖靈數學•統計學叢書. 杜現昆 (漢譯者); 馬晶 (漢譯者). <a href="/wiki/%E4%BA%BA%E6%B0%91%E9%82%AE%E7%94%B5%E5%87%BA%E7%89%88%E7%A4%BE" title="人民邮电出版社">人民郵電出版社</a>. 2009. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/9787115206145" title="Special:网络书源/9787115206145"><span title="国际标准书号">ISBN</span> 9787115206145</a> <span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="连接到中文（中国大陆）网页">（中文（中国大陆））</span>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&rft.au=Sheldon+Axler&rft.btitle=Linear+Algebra+Done+Right&rft.date=2009&rft.genre=book&rft.isbn=9787115206145&rft.pub=%E4%BA%BA%E6%B0%91%E9%83%B5%E9%9B%BB%E5%87%BA%E7%89%88%E7%A4%BE&rft.series=%E5%9C%96%E9%9D%88%E6%95%B8%E5%AD%B8%E2%80%A2%E7%B5%B1%E8%A8%88%E5%AD%B8%E5%8F%A2%E6%9B%B8&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFLax2010" class="citation book"><a href="/wiki/Peter_David_Lax" class="mw-redirect" title="Peter David Lax">Peter D. Lax</a>. <span></span><i>Functional Analysis</i><span></span> [泛函分析]. 圖靈數學·統計學叢書. 侯成軍 (翻譯); 王利廣 (翻譯). 人民郵電出版社. 2010. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-7-115-23174-1" title="Special:网络书源/978-7-115-23174-1"><span title="国际标准书号">ISBN</span> 978-7-115-23174-1</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&rft.au=Peter+D.+Lax&rft.btitle=Functional+Analysis&rft.date=2010&rft.genre=book&rft.isbn=978-7-115-23174-1&rft.pub=%E4%BA%BA%E6%B0%91%E9%83%B5%E9%9B%BB%E5%87%BA%E7%89%88%E7%A4%BE&rft.series=%E5%9C%96%E9%9D%88%E6%95%B8%E5%AD%B8%C2%B7%E7%B5%B1%E8%A8%88%E5%AD%B8%E5%8F%A2%E6%9B%B8&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFFeynman" class="citation book"><a href="/wiki/%E7%90%86%E6%9F%A5%C2%B7%E8%B2%BB%E6%9B%BC" class="mw-redirect" title="理查·費曼">Richard Feynman</a>. <span></span><i>The Feynman Lectures on Physics</i><span></span> [<a href="/wiki/%E8%B2%BB%E6%9B%BC%E7%89%A9%E7%90%86%E5%AD%B8%E8%AC%9B%E7%BE%A9" class="mw-redirect" title="費曼物理學講義">費曼物理學講義</a>] <b>1</b>. <a href="/wiki/%E8%89%BE%E8%BF%AA%E7%94%9F%E9%9F%A6%E6%96%AF%E5%88%A9" title="艾迪生韦斯利">Addison-Wesley</a>. 1999. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0201021165" title="Special:网络书源/978-0201021165"><span title="国际标准书号">ISBN</span> 978-0201021165</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&rft.au=Richard+Feynman&rft.btitle=The+Feynman+Lectures+on+Physics&rft.date=1999&rft.genre=book&rft.isbn=978-0201021165&rft.pub=Addison-Wesley&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="其它參考資料"><span id=".E5.85.B6.E5.AE.83.E5.8F.83.E8.80.83.E8.B3.87.E6.96.99"></span>其它參考資料</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&action=edit&section=22" title="编辑章节：其它參考資料"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Paul_Halmos" class="mw-redirect" title="Paul Halmos">Halmos, Paul R.</a>, <i>Finite-Dimensional Vector Spaces</i>, Springer-Verlag, (1993). <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0387900934" class="internal mw-magiclink-isbn">ISBN 0-387-90093-4</a>.</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 .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf;position:relative}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em;width:auto;padding-left:0.2em;position:absolute;left:1em}.mw-parser-output .navbox .mw-collapsible-toggle{margin-left:0.5em;position:absolute;right:1em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="泛函分析" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84244141"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E6%B3%9B%E5%87%BD%E5%88%86%E6%9E%90" title="Template:泛函分析"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:%E6%B3%9B%E5%87%BD%E5%88%86%E6%9E%90&action=edit&redlink=1" class="new" title="Template talk:泛函分析（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:%E6%B3%9B%E5%87%BD%E5%88%86%E6%9E%90" title="Special:编辑页面/Template:泛函分析"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="泛函分析" style="font-size:110%;margin:0 5em"><a href="/wiki/%E6%B3%9B%E5%87%BD%E5%88%86%E6%9E%90" title="泛函分析">泛函分析</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">集合 / 子集</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/%E7%BB%9D%E5%AF%B9%E5%87%B8%E9%9B%86" title="绝对凸集">绝对凸集</a></li> <li><a href="/wiki/%E5%90%B8%E6%94%B6%E9%9B%86" title="吸收集">吸收集</a></li> <li><a href="/wiki/%E5%B9%B3%E8%A1%A1%E9%9B%86" title="平衡集">平衡集</a></li> <li><span class="ilh-all" data-orig-title="有界集 (拓扑向量空间)" data-lang-code="en" data-lang-name="英语" data-foreign-title="Bounded set (topological vector space)"><span class="ilh-page"><a href="/w/index.php?title=%E6%9C%89%E7%95%8C%E9%9B%86_(%E6%8B%93%E6%89%91%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4)&action=edit&redlink=1" class="new" title="有界集 (拓扑向量空间)（页面不存在）">有界集 (拓扑向量空间)</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Bounded_set_(topological_vector_space)" class="extiw" title="en:Bounded set (topological vector space)"><span lang="en" dir="auto">Bounded set (topological vector space)</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%87%B8%E9%9B%86" title="凸集">凸集</a></li> <li><a href="/wiki/%E5%BE%84%E5%90%91%E9%9B%86" title="径向集">径向集</a></li> <li><a href="/wiki/%E6%98%9F%E5%BD%A2%E5%9F%9F" title="星形域">星形域</a></li> <li><a href="/wiki/%E5%AF%B9%E7%A7%B0%E9%9B%86" title="对称集">对称集</a></li> <li><a href="/wiki/%E5%87%B8%E9%94%A5" title="凸锥">凸锥</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%8B%93%E6%92%B2%E5%90%91%E9%87%8F%E7%A9%BA%E9%96%93" title="拓撲向量空間">拓撲向量空間</a></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/%E5%B7%B4%E6%8B%BF%E8%B5%AB%E7%A9%BA%E9%97%B4" title="巴拿赫空间">巴拿赫空间</a></li> <li><a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%A9%BA%E9%97%B4" title="欧几里得空间">欧几里得空间</a></li> <li><a href="/wiki/%E5%B8%8C%E5%B0%94%E4%BC%AF%E7%89%B9%E7%A9%BA%E9%97%B4" title="希尔伯特空间">希尔伯特空间</a></li> <li><a href="/wiki/%E7%B4%A2%E4%BC%AF%E5%88%97%E5%A4%AB%E7%A9%BA%E9%97%B4" title="索伯列夫空间">索伯列夫空間</a></li> <li><span class="ilh-all" data-orig-title="局部凸" data-lang-code="en" data-lang-name="英语" data-foreign-title="Locally convex topological vector space"><span class="ilh-page"><a href="/w/index.php?title=%E5%B1%80%E9%83%A8%E5%87%B8&action=edit&redlink=1" class="new" title="局部凸（页面不存在）">局部凸</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Locally_convex_topological_vector_space" class="extiw" title="en:Locally convex topological vector space"><span lang="en" dir="auto">Locally convex topological vector space</span></a></span>）</span></span></li> <li><a href="/wiki/%E8%B3%A6%E7%AF%84%E5%90%91%E9%87%8F%E7%A9%BA%E9%96%93" title="賦範向量空間">賦範向量空間</a> （<a href="/wiki/%E8%8C%83%E6%95%B0" title="范数">范数</a>）</li> <li><a href="/wiki/%E6%8B%9F%E8%B5%8B%E8%8C%83%E7%A9%BA%E9%97%B4" title="拟赋范空间">拟赋范空间</a></li> <li><a href="/wiki/%E8%87%AA%E5%8F%8D%E7%A9%BA%E9%97%B4" title="自反空间">自反空间</a></li> <li><span class="ilh-all" data-orig-title="拓扑张量积" data-lang-code="en" data-lang-name="英语" data-foreign-title="Topological tensor product"><span class="ilh-page"><a href="/w/index.php?title=%E6%8B%93%E6%89%91%E5%BC%A0%E9%87%8F%E7%A7%AF&action=edit&redlink=1" class="new" title="拓扑张量积（页面不存在）">拓扑张量积</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Topological_tensor_product" class="extiw" title="en:Topological tensor product"><span lang="en" dir="auto">Topological tensor product</span></a></span>）</span></span> (<a href="/wiki/%E5%B8%8C%E5%B0%94%E4%BC%AF%E7%89%B9%E7%A9%BA%E9%97%B4" title="希尔伯特空间">希尔伯特空间</a>中)</li> <li><a href="/wiki/%E9%80%9F%E9%99%8D%E5%87%BD%E6%95%B0%E7%A9%BA%E9%97%B4" title="速降函数空间">速降函数空间</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">映射拓扑</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/%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/%E7%AE%97%E5%AD%90%E6%8B%93%E6%89%91" title="算子拓扑">算子拓扑</a></li> <li><span class="ilh-all" data-orig-title="弱拓扑" data-lang-code="en" data-lang-name="英语" data-foreign-title="Weak topology"><span class="ilh-page"><a href="/w/index.php?title=%E5%BC%B1%E6%8B%93%E6%89%91&action=edit&redlink=1" class="new" title="弱拓扑（页面不存在）">弱拓扑</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Weak_topology" class="extiw" title="en:Weak topology"><span lang="en" dir="auto">Weak topology</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="强拓扑" data-lang-code="en" data-lang-name="英语" data-foreign-title="Strong topology"><span class="ilh-page"><a href="/w/index.php?title=%E5%BC%BA%E6%8B%93%E6%89%91&action=edit&redlink=1" class="new" title="强拓扑（页面不存在）">强拓扑</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Strong_topology" class="extiw" title="en:Strong topology"><span lang="en" dir="auto">Strong topology</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="拓扑的一致收敛" data-lang-code="en" data-lang-name="英语" data-foreign-title="Topology of uniform convergence"><span class="ilh-page"><a href="/w/index.php?title=%E6%8B%93%E6%89%91%E7%9A%84%E4%B8%80%E8%87%B4%E6%94%B6%E6%95%9B&action=edit&redlink=1" class="new" title="拓扑的一致收敛（页面不存在）">拓扑的一致收敛</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Topology_of_uniform_convergence" class="extiw" title="en:Topology of uniform convergence"><span lang="en" dir="auto">Topology of uniform convergence</span></a></span>）</span></span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">线性算子</a></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/%E5%9F%83%E5%B0%94%E7%B1%B3%E7%89%B9%E4%BC%B4%E9%9A%8F" title="埃尔米特伴随">伴随</a></li> <li><a href="/wiki/%E5%8F%8C%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="双线性映射">双线性</a> <ul><li><a href="/wiki/%E9%9B%99%E7%B7%9A%E6%80%A7%E5%BD%A2%E5%BC%8F" title="雙線性形式">形式</a></li></ul></li> <li><a href="/wiki/%E6%9C%89%E7%95%8C%E7%AE%97%E5%AD%90" title="有界算子">有界</a> / <a href="/wiki/%E6%97%A0%E7%95%8C%E7%AE%97%E5%AD%90" title="无界算子">无界</a></li> <li><a href="/wiki/%E8%BF%9E%E7%BB%AD%E7%BA%BF%E6%80%A7%E7%AE%97%E5%AD%90" title="连续线性算子">连续线性</a></li> <li><a href="/wiki/%E7%B4%A7%E7%AE%97%E5%AD%90" title="紧算子">紧</a></li> <li><span class="ilh-all" data-orig-title="Fredholm算子" data-lang-code="en" data-lang-name="英语" data-foreign-title="Fredholm operator"><span class="ilh-page"><a href="/w/index.php?title=Fredholm%E7%AE%97%E5%AD%90&action=edit&redlink=1" class="new" title="Fredholm算子（页面不存在）">Fredholm</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Fredholm_operator" class="extiw" title="en:Fredholm operator"><span lang="en" dir="auto">Fredholm operator</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%B8%8C%E5%B0%94%E4%BC%AF%E7%89%B9-%E6%96%BD%E5%AF%86%E7%89%B9%E7%AE%97%E5%AD%90" title="希尔伯特-施密特算子">希尔伯特-施密特</a></li> <li><a href="/wiki/%E7%B7%9A%E6%80%A7%E6%B3%9B%E5%87%BD" title="線性泛函">泛函</a></li> <li><a href="/wiki/%E6%AD%A3%E8%A7%84%E7%AE%97%E5%AD%90" title="正规算子">正规</a></li> <li><span class="ilh-all" data-orig-title="核型算子" data-lang-code="en" data-lang-name="英语" data-foreign-title="Nuclear operator"><span class="ilh-page"><a href="/w/index.php?title=%E6%A0%B8%E5%9E%8B%E7%AE%97%E5%AD%90&action=edit&redlink=1" class="new" title="核型算子（页面不存在）">核型</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Nuclear_operator" class="extiw" title="en:Nuclear operator"><span lang="en" dir="auto">Nuclear operator</span></a></span>）</span></span></li> <li><a href="/wiki/%E8%87%AA%E4%BC%B4%E7%AE%97%E5%AD%90" title="自伴算子">自伴</a></li> <li><span class="ilh-all" data-orig-title="严格奇异算子" data-lang-code="en" data-lang-name="英语" data-foreign-title="Strictly singular operator"><span class="ilh-page"><a href="/w/index.php?title=%E4%B8%A5%E6%A0%BC%E5%A5%87%E5%BC%82%E7%AE%97%E5%AD%90&action=edit&redlink=1" class="new" title="严格奇异算子（页面不存在）">严格奇异算子</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Strictly_singular_operator" class="extiw" title="en:Strictly singular operator"><span lang="en" dir="auto">Strictly singular operator</span></a></span>）</span></span></li> <li><a href="/wiki/%E8%BF%B9%E7%B1%BB%E7%AE%97%E5%AD%90" title="迹类算子">迹类</a></li> <li><a href="/wiki/%E6%9C%89%E9%99%90%E7%A7%A9%E7%AE%97%E5%AD%90" title="有限秩算子">有限秩</a></li> <li><span class="ilh-all" data-orig-title="线性映射转置" data-lang-code="en" data-lang-name="英语" data-foreign-title="Transpose of a linear map"><span class="ilh-page"><a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84%E8%BD%AC%E7%BD%AE&action=edit&redlink=1" class="new" title="线性映射转置（页面不存在）">转置</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Transpose_of_a_linear_map" class="extiw" title="en:Transpose of a linear map"><span lang="en" dir="auto">Transpose of a linear map</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%B9%BA%E6%AD%A3%E7%AE%97%E7%AC%A6" title="幺正算符">酉 / 幺正</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">集合运算</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/%E4%BB%A3%E6%95%B0%E5%86%85%E9%83%A8" title="代数内部">代数内部</a></li> <li><a href="/wiki/%E5%86%85%E9%83%A8" title="内部">内部</a></li> <li><a href="/wiki/%E9%96%94%E5%8F%AF%E5%A4%AB%E6%96%AF%E5%9F%BA%E5%92%8C" title="閔可夫斯基和">閔可夫斯基和</a></li> <li><a href="/wiki/%E6%A5%B5%E6%80%A7%E9%9B%86" title="極性集">极性集</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E7%AE%97%E5%AD%90%E7%90%86%E8%AE%BA" title="算子理论">算子理论</a></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/%E5%B7%B4%E6%8B%BF%E8%B5%AB%E4%BB%A3%E6%95%B0" title="巴拿赫代数">巴拿赫代数</a></li> <li><a href="/wiki/C*-%E4%BB%A3%E6%95%B0" title="C*-代数">C*-代数</a></li> <li><a href="/wiki/%E8%B0%B1_(%E6%B3%9B%E5%87%BD%E5%88%86%E6%9E%90)" title="谱 (泛函分析)">谱 (泛函分析)</a> （<a href="/wiki/%E8%B0%B1%E5%8D%8A%E5%BE%84" title="谱半径">谱半径</a>）</li> <li><a href="/wiki/%E8%B0%B1%E7%90%86%E8%AE%BA" title="谱理论">谱理论</a>（<a href="/wiki/%E8%B0%B1%E5%AE%9A%E7%90%86" title="谱定理">谱定理</a></li> <li><a href="/wiki/%E6%8A%95%E5%BD%B1%E5%80%BC%E6%B5%8B%E5%BA%A6" title="投影值测度">投影值测度</a>）</li> <li><a href="/wiki/%E6%9E%81%E5%88%86%E8%A7%A3" title="极分解">极分解</a></li> <li><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></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">定理</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/%E9%98%BF%E5%B0%94%E6%B3%BD%E6%8B%89-%E9%98%BF%E6%96%AF%E7%A7%91%E5%88%A9%E5%AE%9A%E7%90%86" title="阿尔泽拉-阿斯科利定理">阿尔泽拉-阿斯科利定理</a></li> <li><a href="/wiki/%E5%B7%B4%E6%8B%BF%E8%B5%AB-%E9%98%BF%E5%8B%9E%E6%A0%BC%E9%AD%AF%E5%AE%9A%E7%90%86" title="巴拿赫-阿勞格魯定理">巴拿赫-阿勞格魯定理</a></li> <li><span class="ilh-all" data-orig-title="巴拿赫-马祖尔定理" data-lang-code="en" data-lang-name="英语" data-foreign-title="Banach–Mazur theorem"><span class="ilh-page"><a href="/w/index.php?title=%E5%B7%B4%E6%8B%BF%E8%B5%AB-%E9%A9%AC%E7%A5%96%E5%B0%94%E5%AE%9A%E7%90%86&action=edit&redlink=1" class="new" title="巴拿赫-马祖尔定理（页面不存在）">巴拿赫-马祖尔定理</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Mazur_theorem" class="extiw" title="en:Banach–Mazur theorem"><span lang="en" dir="auto">Banach–Mazur theorem</span></a></span>）</span></span></li> <li><a href="/wiki/%E8%B4%9D%E5%B0%94%E7%BA%B2%E5%AE%9A%E7%90%86" title="贝尔纲定理">贝尔纲定理</a></li> <li><a href="/wiki/%E8%B4%9D%E5%A1%9E%E5%B0%94%E4%B8%8D%E7%AD%89%E5%BC%8F" title="贝塞尔不等式">贝塞尔不等式</a></li> <li><a href="/wiki/%E6%9F%AF%E8%A5%BF-%E6%96%BD%E7%93%A6%E8%8C%A8%E4%B8%8D%E7%AD%89%E5%BC%8F" title="柯西-施瓦茨不等式">柯西-施瓦茨不等式</a></li> <li><a href="/wiki/%E9%97%AD%E5%80%BC%E5%9F%9F%E5%AE%9A%E7%90%86" title="闭值域定理">闭值域定理</a></li> <li><a href="/wiki/%E9%96%89%E5%9C%96%E5%83%8F%E5%AE%9A%E7%90%86" title="閉圖像定理">閉圖像定理</a></li> <li><a href="/wiki/%E5%93%88%E6%81%A9-%E5%B7%B4%E6%8B%BF%E8%B5%AB%E5%AE%9A%E7%90%86" title="哈恩-巴拿赫定理">哈恩-巴拿赫定理</a></li> <li><a href="/wiki/%E8%A7%92%E8%B0%B7%E4%B8%8D%E5%8A%A8%E7%82%B9%E5%AE%9A%E7%90%86" title="角谷不动点定理">角谷不动点定理</a></li> <li><a href="/wiki/%E4%B8%8D%E5%8F%98%E5%AD%90%E7%A9%BA%E9%97%B4%E9%97%AE%E9%A2%98" title="不变子空间问题">不变子空间问题</a></li> <li><span class="ilh-all" data-orig-title="Riesz延拓定理" data-lang-code="en" data-lang-name="英语" data-foreign-title="M. Riesz extension theorem"><span class="ilh-page"><a href="/w/index.php?title=Riesz%E5%BB%B6%E6%8B%93%E5%AE%9A%E7%90%86&action=edit&redlink=1" class="new" title="Riesz延拓定理（页面不存在）">Riesz延拓定理</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/M._Riesz_extension_theorem" class="extiw" title="en:M. Riesz extension theorem"><span lang="en" dir="auto">M. Riesz extension theorem</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%BC%80%E6%98%A0%E5%B0%84%E5%AE%9A%E7%90%86" title="开映射定理">开映射定理</a></li> <li><a href="/wiki/%E6%8B%89%E5%85%8B%E6%96%AF-%E7%B1%B3%E7%88%BE%E6%A0%BC%E6%8B%89%E5%A7%86%E5%AE%9A%E7%90%86" title="拉克斯-米爾格拉姆定理">拉克斯-米尔格拉姆定理</a></li> <li><a href="/wiki/%E5%B8%95%E5%A1%9E%E7%93%A6%E5%B0%94%E6%81%92%E7%AD%89%E5%BC%8F" title="帕塞瓦尔恒等式">帕塞瓦尔恒等式</a></li> <li><span class="ilh-all" data-orig-title="肖德尔不动点定理" data-lang-code="en" data-lang-name="英语" data-foreign-title="Schauder fixed point theorem"><span class="ilh-page"><a href="/w/index.php?title=%E8%82%96%E5%BE%B7%E5%B0%94%E4%B8%8D%E5%8A%A8%E7%82%B9%E5%AE%9A%E7%90%86&action=edit&redlink=1" class="new" title="肖德尔不动点定理（页面不存在）">肖德尔不动点定理</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Schauder_fixed_point_theorem" class="extiw" title="en:Schauder fixed point theorem"><span lang="en" dir="auto">Schauder fixed point theorem</span></a></span>）</span></span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">分析</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/%E5%AF%BC%E6%95%B0" title="导数">导数</a> (<a href="/wiki/%E5%BC%97%E9%9B%B7%E6%AD%87%E5%AF%BC%E6%95%B0" title="弗雷歇导数">弗雷歇导数</a></li> <li><a href="/wiki/%E5%8A%A0%E6%89%98%E5%B0%8E%E6%95%B8" title="加托導數">加托导数</a></li> <li><a href="/wiki/%E6%B3%9B%E5%87%BD%E5%AF%BC%E6%95%B0" title="泛函导数">泛函导数</a>)</li> <li><a href="/wiki/%E7%A7%AF%E5%88%86" title="积分">积分</a> (<a href="/wiki/%E5%8D%9A%E8%B5%AB%E7%BA%B3%E7%A7%AF%E5%88%86" title="博赫纳积分">博赫纳积分</a></li> <li><span class="ilh-all" data-orig-title="佩蒂斯积分" data-lang-code="en" data-lang-name="英语" data-foreign-title="Pettis integral"><span class="ilh-page"><a href="/w/index.php?title=%E4%BD%A9%E8%92%82%E6%96%AF%E7%A7%AF%E5%88%86&action=edit&redlink=1" class="new" title="佩蒂斯积分（页面不存在）">佩蒂斯积分</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Pettis_integral" class="extiw" title="en:Pettis integral"><span lang="en" dir="auto">Pettis integral</span></a></span>）</span></span>)</li> <li><a href="/wiki/%E5%87%BD%E6%95%B0%E6%BC%94%E7%AE%97" title="函数演算">函数演算</a> (<a href="/wiki/%E5%85%A8%E7%BA%AF%E5%87%BD%E6%95%B0%E6%BC%94%E7%AE%97" title="全纯函数演算">全纯函数演算</a></li> <li><a href="/wiki/%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0%E6%BC%94%E7%AE%97" title="连续函数演算">连续函数演算</a></li> <li><a href="/wiki/%E5%8D%9A%E9%9B%B7%E5%B0%94%E5%87%BD%E6%95%B0%E6%BC%94%E7%AE%97" title="博雷尔函数演算">博雷尔函数演算</a>)</li> <li><a href="/wiki/%E5%8F%8D%E5%87%BD%E6%95%B0%E5%AE%9A%E7%90%86" title="反函数定理">反函数定理</a></li> <li><a href="/wiki/%E5%90%91%E9%87%8F%E6%B5%8B%E5%BA%A6" title="向量测度">向量测度</a></li> <li><a href="/wiki/%E5%BC%B1%E5%8F%AF%E6%B5%8B%E5%87%BD%E6%95%B0" title="弱可测函数">弱可测函数</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" aria-labelledby="线性代数的相关概念" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84244141"><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%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 class="mw-selflink selflink">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 href="/wiki/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84" title="线性方程组">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 class="mw-selflink selflink">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><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">检索自“<a dir="ltr" href="https://zh.wikipedia.org/w/index.php?title=线性映射&oldid=84918282">https://zh.wikipedia.org/w/index.php?title=线性映射&oldid=84918282</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:%E9%A1%B5%E9%9D%A2%E5%88%86%E7%B1%BB" title="Special:页面分类">分类</a>：<ul><li><a href="/wiki/Category:%E5%87%BD%E6%95%B0" title="Category:函数">函数</a></li><li><a href="/wiki/Category:%E6%8A%BD%E8%B1%A1%E4%BB%A3%E6%95%B0" title="Category:抽象代数">抽象代数</a></li><li><a href="/wiki/Category:%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8" title="Category:線性代數">線性代數</a></li><li><a href="/wiki/Category:%E7%BA%BF%E6%80%A7%E7%AE%97%E5%AD%90" title="Category:线性算子">线性算子</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">隐藏分类：<ul><li><a href="/wiki/Category:%E5%90%AB%E6%9C%89%E8%8B%B1%E8%AA%9E%E7%9A%84%E6%A2%9D%E7%9B%AE" title="Category:含有英語的條目">含有英語的條目</a></li><li><a href="/wiki/Category:%E8%87%AA2016%E5%B9%B46%E6%9C%88%E6%89%A9%E5%85%85%E4%B8%AD%E7%9A%84%E6%9D%A1%E7%9B%AE" title="Category:自2016年6月扩充中的条目">自2016年6月扩充中的条目</a></li><li><a href="/wiki/Category:%E6%89%80%E6%9C%89%E6%89%A9%E5%85%85%E4%B8%AD%E7%9A%84%E6%9D%A1%E7%9B%AE" title="Category:所有扩充中的条目">所有扩充中的条目</a></li><li><a href="/wiki/Category:%E6%8B%92%E7%BB%9D%E5%BD%93%E9%80%89%E9%A6%96%E9%A1%B5%E6%96%B0%E6%9D%A1%E7%9B%AE%E6%8E%A8%E8%8D%90%E6%A0%8F%E7%9B%AE%E7%9A%84%E6%9D%A1%E7%9B%AE" title="Category:拒绝当选首页新条目推荐栏目的条目">拒绝当选首页新条目推荐栏目的条目</a></li><li><a href="/wiki/Category:%E4%BD%BF%E7%94%A8%E5%B0%8F%E5%9E%8B%E8%A8%8A%E6%81%AF%E6%A1%86%E7%9A%84%E9%A0%81%E9%9D%A2" title="Category:使用小型訊息框的頁面">使用小型訊息框的頁面</a></li><li><a href="/wiki/Category:%E4%BD%BF%E7%94%A8ISBN%E9%AD%94%E6%9C%AF%E9%93%BE%E6%8E%A5%E7%9A%84%E9%A1%B5%E9%9D%A2" title="Category:使用ISBN魔术链接的页面">使用ISBN魔术链接的页面</a></li><li><a href="/wiki/Category:%E4%BD%BF%E7%94%A8%E8%BF%87%E6%97%B6%E7%9A%84math%E6%A0%87%E7%AD%BE%E6%A0%BC%E5%BC%8F%E7%9A%84%E9%A1%B5%E9%9D%A2" title="Category:使用过时的math标签格式的页面">使用过时的math标签格式的页面</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> 本页面最后修订于2024年11月10日 (星期日) 10:56。</li> <li id="footer-info-copyright">本站的全部文字在<a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.zh">知识共享署名-相同方式共享 4.0协议</a>之条款下提供，附加条款亦可能应用。（请参阅<a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">使用条款</a>）<br /> Wikipedia®和维基百科标志是<a rel="nofollow" class="external text" href="https://wikimediafoundation.org/zh">维基媒体基金会</a>的注册商标；维基™是维基媒体基金会的商标。<br /> 维基媒体基金会是按美国国內稅收法501(c)(3)登记的<a class="external text" href="https://donate.wikimedia.org/wiki/Special:MyLanguage/Tax_deductibility">非营利慈善机构</a>。<br /></li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">隐私政策</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:%E5%85%B3%E4%BA%8E">关于维基百科</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:%E5%85%8D%E8%B4%A3%E5%A3%B0%E6%98%8E">免责声明</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">行为准则</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">开发者</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/zh.wikipedia.org">统计</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie声明</a></li> <li id="footer-places-mobileview"><a href="//zh.m.wikipedia.org/w/index.php?title=%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">手机版视图</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-65496f48b4-qljwx","wgBackendResponseTime":218,"wgPageParseReport":{"limitreport":{"cputime":"0.881","walltime":"1.171","ppvisitednodes":{"value":2834,"limit":1000000},"postexpandincludesize":{"value":127432,"limit":2097152},"templateargumentsize":{"value":4995,"limit":2097152},"expansiondepth":{"value":8,"limit":100},"expensivefunctioncount":{"value":23,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":40789,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 652.780 1 -total"," 19.78% 129.088 3 Template:Lang-en"," 19.27% 125.817 1 Template:Reflist"," 17.79% 116.101 2 Template:Navbox"," 13.64% 89.046 1 Template:Expand_section"," 13.39% 87.375 1 Template:泛函分析"," 12.78% 83.414 1 Template:Ambox"," 11.86% 77.429 7 Template:Cite_book"," 11.77% 76.863 1 Template:线性代数"," 11.40% 74.406 1 Template:ScienceNavigation"]},"scribunto":{"limitreport-timeusage":{"value":"0.368","limit":"10.000"},"limitreport-memusage":{"value":24685657,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFArtin2010\"] = 1,\n [\"CITEREFAxler2009\"] = 1,\n [\"CITEREFFeynman\"] = 1,\n [\"CITEREFLax2010\"] = 1,\n [\"CITEREFWalter_Rudin1973\"] = 1,\n [\"CITEREFА·Н·柯爾莫哥洛夫，佛明(С._В._Фомин)2006\"] = 1,\n [\"CITEREF李尚志2006\"] = 1,\n}\ntemplate_list = table#1 {\n [\"Cite book\"] = 7,\n [\"DEFAULTSORT:Linear map\"] = 1,\n [\"Expand section\"] = 1,\n [\"Harvnb\"] = 10,\n [\"Lang-en\"] = 3,\n [\"NoteTA\"] = 1,\n [\"Reflist\"] = 1,\n [\"Sfnref\"] = 4,\n [\"泛函分析\"] = 1,\n [\"線性代數\"] = 1,\n [\"線性代數的相關概念\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-api-int.codfw.main-648bd44df8-bxkj4","timestamp":"20241116023315","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"\u7ebf\u6027\u6620\u5c04","url":"https:\/\/zh.wikipedia.org\/wiki\/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84","sameAs":"http:\/\/www.wikidata.org\/entity\/Q207643","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q207643","author":{"@type":"Organization","name":"\u7ef4\u57fa\u5a92\u4f53\u9879\u76ee\u8d21\u732e\u8005"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2007-06-18T10:08:21Z","dateModified":"2024-11-10T10:56:56Z","headline":"\u4fdd\u7559\u52a0\u6cd5\u548c\u6a19\u91cf\u4e58\u6cd5\u904b\u7b97\u7684\u6620\u5c04"}</script> </body> </html>

CINXE.COM

线性映射 - 维基百科，自由的百科全书