CINXE.COM

<!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="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Linear span - Wikipedia</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(/(?:^|; )enwikimwclientpreferences=([^;]+)/);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":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"6a3f3d96-268a-465c-acb4-50a12ac3bb49","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Linear_span","wgTitle":"Linear span","wgCurRevisionId":1251730161,"wgRevisionId":1251730161,"wgArticleId":56353,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","Abstract algebra","Linear algebra"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Linear_span","wgRelevantArticleId":56353,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{ "levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q209812","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles" :"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.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","mediawiki.page.media","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","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=en&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=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&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 property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/7/70/Basis_for_a_plane.svg/1200px-Basis_for_a_plane.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="668"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/7/70/Basis_for_a_plane.svg/800px-Basis_for_a_plane.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="445"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/7/70/Basis_for_a_plane.svg/640px-Basis_for_a_plane.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="356"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Linear span - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Linear_span"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Linear_span&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 (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Linear_span"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&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-Linear_span rootpage-Linear_span skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</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="Site"> <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="Main menu" > <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">Main menu</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">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" 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="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </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:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</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="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [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:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <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="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <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="Appearance" > <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">Appearance</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/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</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:CreateAccount&returnto=Linear+span" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</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:UserLogin&returnto=Linear+span" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</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="Log in and more options" > <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="Personal tools" > <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">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <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/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Linear+span" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Linear+span" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</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"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</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:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</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="Site"> <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="Contents" 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">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</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">(Top)</div> </a> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definition</span> </div> </a> <ul id="toc-Definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theorems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Theorems"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Theorems</span> </div> </a> <button aria-controls="toc-Theorems-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>Toggle Theorems subsection</span> </button> <ul id="toc-Theorems-sublist" class="vector-toc-list"> <li id="toc-Equivalence_of_definitions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalence_of_definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Equivalence of definitions</span> </div> </a> <ul id="toc-Equivalence_of_definitions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Size_of_spanning_set_is_at_least_size_of_linearly_independent_set" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Size_of_spanning_set_is_at_least_size_of_linearly_independent_set"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Size of spanning set is at least size of linearly independent set</span> </div> </a> <ul id="toc-Size_of_spanning_set_is_at_least_size_of_linearly_independent_set-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spanning_set_can_be_reduced_to_a_basis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spanning_set_can_be_reduced_to_a_basis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Spanning set can be reduced to a basis</span> </div> </a> <ul id="toc-Spanning_set_can_be_reduced_to_a_basis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Generalizations</span> </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Closed_linear_span_(functional_analysis)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Closed_linear_span_(functional_analysis)"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Closed linear span (functional analysis)</span> </div> </a> <button aria-controls="toc-Closed_linear_span_(functional_analysis)-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>Toggle Closed linear span (functional analysis) subsection</span> </button> <ul id="toc-Closed_linear_span_(functional_analysis)-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_useful_lemma" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_useful_lemma"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>A useful lemma</span> </div> </a> <ul id="toc-A_useful_lemma-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Sources</span> </div> </a> <button aria-controls="toc-Sources-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>Toggle Sources subsection</span> </button> <ul id="toc-Sources-sublist" class="vector-toc-list"> <li id="toc-Textbooks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Textbooks"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Textbooks</span> </div> </a> <ul id="toc-Textbooks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Web" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Web"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Web</span> </div> </a> <ul id="toc-Web-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" 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="Toggle the table of contents" > <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">Toggle the table of contents</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">Linear span</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="Go to an article in another language. Available in 25 languages" > <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-25" 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">25 languages</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/%D9%85%D8%AF%D9%89_%D8%AE%D8%B7%D9%8A" title="مدى خطي – Arabic" lang="ar" hreflang="ar" data-title="مدى خطي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/S%C3%B2a%E2%81%BF-s%C3%A8ng_th%C3%AD-khui" title="Sòaⁿ-sèng thí-khui – Minnan" lang="nan" hreflang="nan" data-title="Sòaⁿ-sèng thí-khui" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_vectorial_generat" title="Espai vectorial generat – Catalan" lang="ca" hreflang="ca" data-title="Espai vectorial generat" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Line%C3%A1rn%C3%AD_obal" title="Lineární obal – Czech" lang="cs" hreflang="cs" data-title="Lineární obal" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Lineare_H%C3%BClle" title="Lineare Hülle – German" lang="de" hreflang="de" data-title="Lineare Hülle" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sistema_generador" title="Sistema generador – Spanish" lang="es" hreflang="es" data-title="Sistema generador" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%BE%D9%88%D8%B4%D8%B4_%D8%AE%D8%B7%DB%8C" title="پوشش خطی – Persian" lang="fa" hreflang="fa" data-title="پوشش خطی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Sous-espace_vectoriel_engendr%C3%A9" title="Sous-espace vectoriel engendré – French" lang="fr" hreflang="fr" data-title="Sous-espace vectoriel engendré" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</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_%EC%83%9D%EC%84%B1" title="선형 생성 – Korean" lang="ko" hreflang="ko" data-title="선형 생성" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Rentang_linear" title="Rentang linear – Indonesian" lang="id" hreflang="id" data-title="Rentang linear" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" 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_sp%C3%B6nn" title="Línuleg spönn – Icelandic" lang="is" hreflang="is" data-title="Línuleg spönn" data-language-autonym="Íslenska" data-language-local-name="Icelandic" 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/Copertura_lineare" title="Copertura lineare – Italian" lang="it" hreflang="it" data-title="Copertura lineare" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%91%D7%95%D7%A6%D7%94_%D7%A4%D7%95%D7%A8%D7%A9%D7%AA" title="קבוצה פורשת – Hebrew" lang="he" hreflang="he" data-title="קבוצה פורשת" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Gener%C3%A1torrendszer_(line%C3%A1ris_algebra)" title="Generátorrendszer (lineáris algebra) – Hungarian" lang="hu" hreflang="hu" data-title="Generátorrendszer (lineáris algebra)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lineair_omhulsel" title="Lineair omhulsel – Dutch" lang="nl" hreflang="nl" data-title="Lineair omhulsel" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</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%8C%85" title="線型包 – Japanese" lang="ja" hreflang="ja" data-title="線型包" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_vectorial_gerado" title="Espaço vectorial gerado – Portuguese" lang="pt" hreflang="pt" data-title="Espaço vectorial gerado" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%B0%D1%8F_%D0%BE%D0%B1%D0%BE%D0%BB%D0%BE%D1%87%D0%BA%D0%B0" title="Линейная оболочка – Russian" lang="ru" hreflang="ru" data-title="Линейная оболочка" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Spani_linear" title="Spani linear – Albanian" lang="sq" hreflang="sq" data-title="Spani linear" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Linearna_ogrinja%C4%8Da" title="Linearna ogrinjača – Slovenian" lang="sl" hreflang="sl" data-title="Linearna ogrinjača" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Linj%C3%A4rt_h%C3%B6lje" title="Linjärt hölje – Swedish" lang="sv" hreflang="sv" data-title="Linjärt hölje" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Do%C4%9Frusal_germe" title="Doğrusal germe – Turkish" lang="tr" hreflang="tr" data-title="Doğrusal germe" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk badge-Q70893996 mw-list-item" title=""><a href="https://uk.wikipedia.org/wiki/%D0%9B%D1%96%D0%BD%D1%96%D0%B9%D0%BD%D0%B0_%D0%BE%D0%B1%D0%BE%D0%BB%D0%BE%D0%BD%D0%BA%D0%B0" title="Лінійна оболонка – Ukrainian" lang="uk" hreflang="uk" data-title="Лінійна оболонка" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Span_tuy%E1%BA%BFn_t%C3%ADnh" title="Span tuyến tính – Vietnamese" lang="vi" hreflang="vi" data-title="Span tuyến tính" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%BA%BF%E6%80%A7%E7%94%9F%E6%88%90%E7%A9%BA%E9%97%B4" title="线性生成空间 – Chinese" lang="zh" hreflang="zh" data-title="线性生成空间" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q209812#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</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="Namespaces"> <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/Linear_span" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Linear_span" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <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">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <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/Linear_span"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Linear_span&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Linear_span&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <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="Tools" > <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">Tools</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">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </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/Linear_span"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Linear_span&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Linear_span&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Linear_span" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Linear_span" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Linear_span&oldid=1251730161" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Linear_span&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Linear_span&id=1251730161&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FLinear_span"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FLinear_span"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Linear_span&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Linear_span&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</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"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q209812" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</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="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <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">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</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> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</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="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">In linear algebra, generated subspace</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Basis_for_a_plane.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Basis_for_a_plane.svg/280px-Basis_for_a_plane.svg.png" decoding="async" width="280" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Basis_for_a_plane.svg/420px-Basis_for_a_plane.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Basis_for_a_plane.svg/560px-Basis_for_a_plane.svg.png 2x" data-file-width="512" data-file-height="285" /></a><figcaption>The cross-hatched plane is the linear span of <b>u</b> and <b>v</b> in both <b>R</b><sup>2</sup> and <b>R</b><sup>3</sup>.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>linear span</b> (also called the <b>linear hull</b><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> or just <b>span</b>) of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> of elements of a <a href="/wiki/Vector_space" title="Vector space">vector space</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 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> is the smallest <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> of <span class="mwe-math-element"><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> that contains <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23bbb1f0f6ebdfa78b4fed06049640f7386bb44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.146ex; height:2.176ex;" alt="{\displaystyle S.}"></span> It is the set of all finite <a href="/wiki/Linear_combination" title="Linear combination">linear combinations</a> of the elements of <span class="texhtml mvar" style="font-style:italic;">S</span>,<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> and the intersection of all linear subspaces that contain <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23bbb1f0f6ebdfa78b4fed06049640f7386bb44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.146ex; height:2.176ex;" alt="{\displaystyle S.}"></span> It often denoted <span class="texhtml">span(<i>S</i>)</span><sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle S\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>S</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle S\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e4ebebb1ab0a24cd75f3c2cf013bb4a63ed6709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.955ex; height:2.843ex;" alt="{\displaystyle \langle S\rangle .}"></span> </p><p>For example, in <a href="/wiki/Geometry" title="Geometry">geometry</a>, two <a href="/wiki/Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> <a href="/wiki/Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">vectors</a> span a <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a>. </p><p>To express that a vector space <span class="texhtml mvar" style="font-style:italic;">V</span> is a linear span of a subset <span class="texhtml mvar" style="font-style:italic;">S</span>, one commonly uses one of the following phrases: <span class="texhtml mvar" style="font-style:italic;">S</span> spans <span class="texhtml mvar" style="font-style:italic;">V</span>; <span class="texhtml mvar" style="font-style:italic;">S</span> is a <b>spanning set</b> of <span class="texhtml mvar" style="font-style:italic;">V</span>; <span class="texhtml mvar" style="font-style:italic;">V</span> is spanned or <a href="/wiki/Generator_(mathematics)" title="Generator (mathematics)">generated</a> by <span class="texhtml mvar" style="font-style:italic;">S</span>; <span class="texhtml mvar" style="font-style:italic;">S</span> is a generator set or a generating set of <span class="texhtml mvar" style="font-style:italic;">V</span>. </p><p>Spans can be generalized to many <a href="/wiki/Mathematical_structure" title="Mathematical structure">mathematical structures</a>, in which case, the smallest substructure containing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> is generally called the substructure <i>generated</i> by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23bbb1f0f6ebdfa78b4fed06049640f7386bb44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.146ex; height:2.176ex;" alt="{\displaystyle S.}"></span> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_span&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a <a href="/wiki/Vector_space" title="Vector space">vector space</a> <span class="texhtml mvar" style="font-style:italic;">V</span> over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <span class="texhtml mvar" style="font-style:italic;">K</span>, the span of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <span class="texhtml mvar" style="font-style:italic;">S</span> of vectors (not necessarily finite) is defined to be the intersection <span class="texhtml mvar" style="font-style:italic;">W</span> of all <a href="/wiki/Linear_subspace" title="Linear subspace">subspaces</a> of <span class="texhtml mvar" style="font-style:italic;">V</span> that contain <span class="texhtml mvar" style="font-style:italic;">S</span>. It is thus the smallest (for <a href="/wiki/Set_inclusion" class="mw-redirect" title="Set inclusion">set inclusion</a>) subspace containing <span class="texhtml mvar" style="font-style:italic;">W</span>. It is referred to as the subspace <i>spanned by</i> <span class="texhtml mvar" style="font-style:italic;">S</span>, or by the vectors in <span class="texhtml mvar" style="font-style:italic;">S</span>. Conversely, <span class="texhtml mvar" style="font-style:italic;">S</span> is called a <i>spanning set</i> of <span class="texhtml mvar" style="font-style:italic;">W</span>, and we say that <span class="texhtml mvar" style="font-style:italic;">S</span> <i>spans</i> <span class="texhtml mvar" style="font-style:italic;">W</span>. </p><p>It follows from this definition that the span of <span class="texhtml mvar" style="font-style:italic;">S</span> is the set of all finite <a href="/wiki/Linear_combinations" class="mw-redirect" title="Linear combinations">linear combinations</a> of elements (vectors) of <span class="texhtml mvar" style="font-style:italic;">S</span>, and can be defined as such.<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-:02_5-0" class="reference"><a href="#cite_note-:02-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> That is, <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 \operatorname {span} (S)={\biggl \{}\lambda _{1}\mathbf {v} _{1}+\lambda _{2}\mathbf {v} _{2}+\cdots +\lambda _{n}\mathbf {v} _{n}\mid n\in \mathbb {N} ,\;\mathbf {v} _{1},...\mathbf {v} _{n}\in S,\;\lambda _{1},...\lambda _{n}\in K{\biggr \}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>span</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">{</mo> </mrow> </mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mi>S</mi> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {span} (S)={\biggl \{}\lambda _{1}\mathbf {v} _{1}+\lambda _{2}\mathbf {v} _{2}+\cdots +\lambda _{n}\mathbf {v} _{n}\mid n\in \mathbb {N} ,\;\mathbf {v} _{1},...\mathbf {v} _{n}\in S,\;\lambda _{1},...\lambda _{n}\in K{\biggr \}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2e43654c3b22ce5c3ede260cfe383ca858030a9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:79.559ex; height:6.176ex;" alt="{\displaystyle \operatorname {span} (S)={\biggl \{}\lambda _{1}\mathbf {v} _{1}+\lambda _{2}\mathbf {v} _{2}+\cdots +\lambda _{n}\mathbf {v} _{n}\mid n\in \mathbb {N} ,\;\mathbf {v} _{1},...\mathbf {v} _{n}\in S,\;\lambda _{1},...\lambda _{n}\in K{\biggr \}}}"></span> </p><p>When <span class="texhtml mvar" style="font-style:italic;">S</span> is <a href="/wiki/Empty_set" title="Empty set">empty</a>, the only possibility is <span class="texhtml"><i>n</i> = 0</span>, and the previous expression for <span class="mwe-math-element"><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 {span} (S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>span</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {span} (S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/185bcd5416275cd32c9de5427eb20e1b55153f54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.972ex; height:2.843ex;" alt="{\displaystyle \operatorname {span} (S)}"></span> reduces to the <a href="/wiki/Empty_sum" title="Empty sum">empty sum</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> The standard convention for the empty sum implies thus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{span}}(\emptyset )=\{\mathbf {0} \},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>span</mtext> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∅</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{span}}(\emptyset )=\{\mathbf {0} \},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c12ffe204597126111e251ad8f609ff5ae326c84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.042ex; height:2.843ex;" alt="{\displaystyle {\text{span}}(\emptyset )=\{\mathbf {0} \},}"></span> a property that is immediate with the other definitions. However, many introductory textbooks simply include this fact as part of the definition. </p><p>When <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\{\mathbf {v} _{1},\ldots ,\mathbf {v} _{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\{\mathbf {v} _{1},\ldots ,\mathbf {v} _{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0ac0bae489d45dac001f26c0c52c06b15f26050" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.195ex; height:2.843ex;" alt="{\displaystyle S=\{\mathbf {v} _{1},\ldots ,\mathbf {v} _{n}\}}"></span> is <a href="/wiki/Finite_set" title="Finite set">finite</a>, one has <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 \operatorname {span} (S)=\{\lambda _{1}\mathbf {v} _{1}+\lambda _{2}\mathbf {v} _{2}+\cdots +\lambda _{n}\mathbf {v} _{n}\mid \lambda _{1},...\lambda _{n}\in K\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>span</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mi>K</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {span} (S)=\{\lambda _{1}\mathbf {v} _{1}+\lambda _{2}\mathbf {v} _{2}+\cdots +\lambda _{n}\mathbf {v} _{n}\mid \lambda _{1},...\lambda _{n}\in K\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f26ec80a76de02dfd4afa480939109354f55799e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.555ex; height:2.843ex;" alt="{\displaystyle \operatorname {span} (S)=\{\lambda _{1}\mathbf {v} _{1}+\lambda _{2}\mathbf {v} _{2}+\cdots +\lambda _{n}\mathbf {v} _{n}\mid \lambda _{1},...\lambda _{n}\in K\}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_span&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Real_number" title="Real number">real</a> vector space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span> has {(−1, 0, 0), (0, 1, 0), (0, 0, 1)} as a spanning set. This particular spanning set is also a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a>. If (−1, 0, 0) were replaced by (1, 0, 0), it would also form the <a href="/wiki/Standard_basis" title="Standard basis">canonical basis</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span>. </p><p>Another spanning set for the same space is given by {(1, 2, 3), (0, 1, 2), (−1, <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>, 3), (1, 1, 1)}, but this set is not a basis, because it is <a href="/wiki/Linear_dependency" class="mw-redirect" title="Linear dependency">linearly dependent</a>. </p><p>The set <span class="texhtml">{(1, 0, 0), (0, 1, 0), (1, 1, 0)</span>} is not a spanning set of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span>, since its span is the space of all vectors in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span> whose last component is zero. That space is also spanned by the set {(1, 0, 0), (0, 1, 0)}, as (1, 1, 0) is a linear combination of (1, 0, 0) and (0, 1, 0). Thus, the spanned space is not <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b00b2b4fd27c2cbffa02df568472f77b194a6db9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}.}"></span> It can be identified with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> by removing the third components equal to zero. </p><p>The empty set is a spanning set of {(0, 0, 0)}, since the empty set is a subset of all possible vector spaces in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span>, and {(0, 0, 0)} is the intersection of all of these vector spaces. </p><p>The set of <a href="/wiki/Monomial" title="Monomial">monomials</a> <span class="texhtml mvar" style="font-style:italic;">x<sup>n</sup></span>, where <span class="texhtml mvar" style="font-style:italic;">n</span> is a non-negative integer, spans the space of <a href="/wiki/Polynomial" title="Polynomial">polynomials</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Theorems">Theorems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_span&action=edit&section=3" title="Edit section: Theorems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Equivalence_of_definitions">Equivalence of definitions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_span&action=edit&section=4" title="Edit section: Equivalence of definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The set of all linear combinations of a subset <span class="texhtml mvar" style="font-style:italic;">S</span> of <span class="texhtml mvar" style="font-style:italic;">V</span>, a vector space over <span class="texhtml mvar" style="font-style:italic;">K</span>, is the smallest linear subspace of <span class="texhtml mvar" style="font-style:italic;">V</span> containing <span class="texhtml mvar" style="font-style:italic;">S</span>. </p> <dl><dd><i>Proof.</i> We first prove that <span class="texhtml">span <i>S</i></span> is a subspace of <span class="texhtml mvar" style="font-style:italic;">V</span>. Since <span class="texhtml mvar" style="font-style:italic;">S</span> is a subset of <span class="texhtml mvar" style="font-style:italic;">V</span>, we only need to prove the existence of a zero vector <span class="texhtml"><b>0</b></span> in <span class="texhtml">span <i>S</i></span>, that <span class="texhtml">span <i>S</i></span> is closed under addition, and that <span class="texhtml">span <i>S</i></span> is closed under scalar multiplication. Letting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\{\mathbf {v} _{1},\mathbf {v} _{2},\ldots ,\mathbf {v} _{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\{\mathbf {v} _{1},\mathbf {v} _{2},\ldots ,\mathbf {v} _{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14fe8d8f5069f14591e002c65c22e1d4acd21cc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.695ex; height:2.843ex;" alt="{\displaystyle S=\{\mathbf {v} _{1},\mathbf {v} _{2},\ldots ,\mathbf {v} _{n}\}}"></span>, it is trivial that the zero vector of <span class="texhtml mvar" style="font-style:italic;">V</span> exists in <span class="texhtml">span <i>S</i></span>, since <span class="mwe-math-element"><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 {0} =0\mathbf {v} _{1}+0\mathbf {v} _{2}+\cdots +0\mathbf {v} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>=</mo> <mn>0</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>0</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mn>0</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} =0\mathbf {v} _{1}+0\mathbf {v} _{2}+\cdots +0\mathbf {v} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c12559ade30676d8f91f0ed5280081ab6397ddb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.727ex; height:2.509ex;" alt="{\displaystyle \mathbf {0} =0\mathbf {v} _{1}+0\mathbf {v} _{2}+\cdots +0\mathbf {v} _{n}}"></span>. Adding together two linear combinations of <span class="texhtml mvar" style="font-style:italic;">S</span> also produces a linear combination of <span class="texhtml mvar" style="font-style:italic;">S</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda _{1}\mathbf {v} _{1}+\cdots +\lambda _{n}\mathbf {v} _{n})+(\mu _{1}\mathbf {v} _{1}+\cdots +\mu _{n}\mathbf {v} _{n})=(\lambda _{1}+\mu _{1})\mathbf {v} _{1}+\cdots +(\lambda _{n}+\mu _{n})\mathbf {v} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda _{1}\mathbf {v} _{1}+\cdots +\lambda _{n}\mathbf {v} _{n})+(\mu _{1}\mathbf {v} _{1}+\cdots +\mu _{n}\mathbf {v} _{n})=(\lambda _{1}+\mu _{1})\mathbf {v} _{1}+\cdots +(\lambda _{n}+\mu _{n})\mathbf {v} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce23ab9c2b3f2015d13932c285c01b7ffd7df4dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:79.471ex; height:2.843ex;" alt="{\displaystyle (\lambda _{1}\mathbf {v} _{1}+\cdots +\lambda _{n}\mathbf {v} _{n})+(\mu _{1}\mathbf {v} _{1}+\cdots +\mu _{n}\mathbf {v} _{n})=(\lambda _{1}+\mu _{1})\mathbf {v} _{1}+\cdots +(\lambda _{n}+\mu _{n})\mathbf {v} _{n}}"></span>, where all <span class="mwe-math-element"><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 _{i},\mu _{i}\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i},\mu _{i}\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/488e9dbe508cac2327c2387f0c3760e95625fc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.297ex; height:2.676ex;" alt="{\displaystyle \lambda _{i},\mu _{i}\in K}"></span>, and multiplying a linear combination of <span class="texhtml mvar" style="font-style:italic;">S</span> by a scalar <span class="mwe-math-element"><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\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e894da36b4db8c32e35962d3950750ad556f46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle c\in K}"></span> will produce another linear combination of <span class="texhtml mvar" style="font-style:italic;">S</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(\lambda _{1}\mathbf {v} _{1}+\cdots +\lambda _{n}\mathbf {v} _{n})=c\lambda _{1}\mathbf {v} _{1}+\cdots +c\lambda _{n}\mathbf {v} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>c</mi> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(\lambda _{1}\mathbf {v} _{1}+\cdots +\lambda _{n}\mathbf {v} _{n})=c\lambda _{1}\mathbf {v} _{1}+\cdots +c\lambda _{n}\mathbf {v} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f898d3d824ab7a733ee5bfdffc81a4fe59c39b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.892ex; height:2.843ex;" alt="{\displaystyle c(\lambda _{1}\mathbf {v} _{1}+\cdots +\lambda _{n}\mathbf {v} _{n})=c\lambda _{1}\mathbf {v} _{1}+\cdots +c\lambda _{n}\mathbf {v} _{n}}"></span>. Thus <span class="texhtml">span <i>S</i></span> is a subspace of <span class="texhtml mvar" style="font-style:italic;">V</span>.</dd></dl> <dl><dd>Suppose that <span class="texhtml mvar" style="font-style:italic;">W</span> is a linear subspace of <span class="texhtml mvar" style="font-style:italic;">V</span> containing <span class="texhtml mvar" style="font-style:italic;">S</span>. It follows that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq \operatorname {span} S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>span</mi> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq \operatorname {span} S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d99fbb3d10477160b46714a8d2186ef78ae923" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.148ex; height:2.509ex;" alt="{\displaystyle S\subseteq \operatorname {span} S}"></span>, since every <span class="texhtml"><b>v</b><sub><i>i</i></sub></span> is a linear combination of <span class="texhtml mvar" style="font-style:italic;">S</span> (trivially). Since <span class="texhtml mvar" style="font-style:italic;">W</span> is closed under addition and scalar multiplication, then every linear combination <span class="mwe-math-element"><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}\mathbf {v} _{1}+\cdots +\lambda _{n}\mathbf {v} _{n}}"> <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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1}\mathbf {v} _{1}+\cdots +\lambda _{n}\mathbf {v} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d82e7f7354e332d33b0aa3ac6b4c9295b3232d72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.482ex; height:2.509ex;" alt="{\displaystyle \lambda _{1}\mathbf {v} _{1}+\cdots +\lambda _{n}\mathbf {v} _{n}}"></span> must be contained in <span class="texhtml mvar" style="font-style:italic;">W</span>. Thus, <span class="texhtml">span <i>S</i></span> is contained in every subspace of <span class="texhtml mvar" style="font-style:italic;">V</span> containing <span class="texhtml mvar" style="font-style:italic;">S</span>, and the intersection of all such subspaces, or the smallest such subspace, is equal to the set of all linear combinations of <span class="texhtml mvar" style="font-style:italic;">S</span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Size_of_spanning_set_is_at_least_size_of_linearly_independent_set">Size of spanning set is at least size of linearly independent set</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_span&action=edit&section=5" title="Edit section: Size of spanning set is at least size of linearly independent set"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every spanning set <span class="texhtml mvar" style="font-style:italic;">S</span> of a vector space <span class="texhtml mvar" style="font-style:italic;">V</span> must contain at least as many elements as any <a href="/wiki/Linear_independence" title="Linear independence">linearly independent</a> set of vectors from <span class="texhtml mvar" style="font-style:italic;">V</span>. </p> <dl><dd><i>Proof.</i> Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\{\mathbf {v} _{1},\ldots ,\mathbf {v} _{m}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\{\mathbf {v} _{1},\ldots ,\mathbf {v} _{m}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f4aa1cb6fa7013a445c6c7a9f1df2d099a08e09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.652ex; height:2.843ex;" alt="{\displaystyle S=\{\mathbf {v} _{1},\ldots ,\mathbf {v} _{m}\}}"></span> be a spanning set and <span class="mwe-math-element"><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=\{\mathbf {w} _{1},\ldots ,\mathbf {w} _{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\{\mathbf {w} _{1},\ldots ,\mathbf {w} _{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f9c642ba4d1fa1a15b87b90c97283802eb6e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.172ex; height:2.843ex;" alt="{\displaystyle W=\{\mathbf {w} _{1},\ldots ,\mathbf {w} _{n}\}}"></span> be a linearly independent set of vectors from <span class="texhtml mvar" style="font-style:italic;">V</span>. We want to show that <span class="mwe-math-element"><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\geq 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\geq n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b02f25e62da7fe3162ac80446437cdc1c0fd341" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.534ex; height:2.176ex;" alt="{\displaystyle m\geq n}"></span>.</dd></dl> <dl><dd>Since <span class="texhtml mvar" style="font-style:italic;">S</span> spans <span class="texhtml mvar" style="font-style:italic;">V</span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\cup \{\mathbf {w} _{1}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\cup \{\mathbf {w} _{1}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f391dc56ac54d7844cbecceef21af47d54c72f01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.392ex; height:2.843ex;" alt="{\displaystyle S\cup \{\mathbf {w} _{1}\}}"></span> must also span <span class="texhtml mvar" style="font-style:italic;">V</span>, and <span class="mwe-math-element"><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 {w} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/856752f8af2657e2d348dfd7f2d9d86d4abad771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.009ex;" alt="{\displaystyle \mathbf {w} _{1}}"></span> must be a linear combination of <span class="texhtml mvar" style="font-style:italic;">S</span>. Thus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\cup \{\mathbf {w} _{1}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\cup \{\mathbf {w} _{1}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f391dc56ac54d7844cbecceef21af47d54c72f01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.392ex; height:2.843ex;" alt="{\displaystyle S\cup \{\mathbf {w} _{1}\}}"></span> is linearly dependent, and we can remove one vector from <span class="texhtml mvar" style="font-style:italic;">S</span> that is a linear combination of the other elements. This vector cannot be any of the <span class="texhtml"><b>w</b><sub><i>i</i></sub></span>, since <span class="texhtml mvar" style="font-style:italic;">W</span> is linearly independent. The resulting set is <span class="mwe-math-element"><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 {w} _{1},\mathbf {v} _{1},\ldots ,\mathbf {v} _{i-1},\mathbf {v} _{i+1},\ldots ,\mathbf {v} _{m}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\mathbf {w} _{1},\mathbf {v} _{1},\ldots ,\mathbf {v} _{i-1},\mathbf {v} _{i+1},\ldots ,\mathbf {v} _{m}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d47eba76f1f2956766fa74fdd7c26046f02cb5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.908ex; height:2.843ex;" alt="{\displaystyle \{\mathbf {w} _{1},\mathbf {v} _{1},\ldots ,\mathbf {v} _{i-1},\mathbf {v} _{i+1},\ldots ,\mathbf {v} _{m}\}}"></span>, which is a spanning set of <span class="texhtml mvar" style="font-style:italic;">V</span>. We repeat this step <span class="texhtml mvar" style="font-style:italic;">n</span> times, where the resulting set after the <span class="texhtml mvar" style="font-style:italic;">p</span>th step is the union of <span class="mwe-math-element"><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 {w} _{1},\ldots ,\mathbf {w} _{p}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\mathbf {w} _{1},\ldots ,\mathbf {w} _{p}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f0eeb4c56fcc9b8d5793997d7a9447990bf330" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.479ex; height:3.009ex;" alt="{\displaystyle \{\mathbf {w} _{1},\ldots ,\mathbf {w} _{p}\}}"></span> and <span class="texhtml mvar" style="font-style:italic;">m - p</span> vectors of <span class="texhtml mvar" style="font-style:italic;">S</span>.</dd></dl> <dl><dd>It is ensured until the <span class="texhtml mvar" style="font-style:italic;">n</span>th step that there will always be some <span class="texhtml"><b>v</b><sub><i>i</i></sub></span> to remove out of <span class="texhtml mvar" style="font-style:italic;">S</span> for every adjoint of <span class="texhtml"><b>v</b></span>, and thus there are at least as many <span class="texhtml"><b>v</b><sub><i>i</i></sub></span>'s as there are <span class="texhtml"><b>w</b><sub><i>i</i></sub></span>'s—i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\geq 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\geq n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b02f25e62da7fe3162ac80446437cdc1c0fd341" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.534ex; height:2.176ex;" alt="{\displaystyle m\geq n}"></span>. To verify this, we assume by way of contradiction that <span class="mwe-math-element"><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<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<n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/490c01b0cb770144f28afd17bb5fef277daf6f38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.843ex;" alt="{\displaystyle m<n}"></span>. Then, at the <span class="texhtml mvar" style="font-style:italic;">m</span>th step, we have the set <span class="mwe-math-element"><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 {w} _{1},\ldots ,\mathbf {w} _{m}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\mathbf {w} _{1},\ldots ,\mathbf {w} _{m}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d540f5ca6b88d7f50b2a679f4f2b500c18d6fd99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.095ex; height:2.843ex;" alt="{\displaystyle \{\mathbf {w} _{1},\ldots ,\mathbf {w} _{m}\}}"></span> and we can adjoin another vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} _{m+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} _{m+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/500f3633021d8cec54e78aa689136175b0c343d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.707ex; height:2.009ex;" alt="{\displaystyle \mathbf {w} _{m+1}}"></span>. But, since <span class="mwe-math-element"><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 {w} _{1},\ldots ,\mathbf {w} _{m}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\mathbf {w} _{1},\ldots ,\mathbf {w} _{m}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d540f5ca6b88d7f50b2a679f4f2b500c18d6fd99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.095ex; height:2.843ex;" alt="{\displaystyle \{\mathbf {w} _{1},\ldots ,\mathbf {w} _{m}\}}"></span> is a spanning set of <span class="texhtml mvar" style="font-style:italic;">V</span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} _{m+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} _{m+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/500f3633021d8cec54e78aa689136175b0c343d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.707ex; height:2.009ex;" alt="{\displaystyle \mathbf {w} _{m+1}}"></span> is a linear combination of <span class="mwe-math-element"><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 {w} _{1},\ldots ,\mathbf {w} _{m}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\mathbf {w} _{1},\ldots ,\mathbf {w} _{m}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d540f5ca6b88d7f50b2a679f4f2b500c18d6fd99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.095ex; height:2.843ex;" alt="{\displaystyle \{\mathbf {w} _{1},\ldots ,\mathbf {w} _{m}\}}"></span>. This is a contradiction, since <span class="texhtml mvar" style="font-style:italic;">W</span> is linearly independent.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Spanning_set_can_be_reduced_to_a_basis">Spanning set can be reduced to a basis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_span&action=edit&section=6" title="Edit section: Spanning set can be reduced to a basis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml mvar" style="font-style:italic;">V</span> be a finite-dimensional vector space. Any set of vectors that spans <span class="texhtml mvar" style="font-style:italic;">V</span> can be reduced to a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> for <span class="texhtml mvar" style="font-style:italic;">V</span>, by discarding vectors if necessary (i.e. if there are linearly dependent vectors in the set). If the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> holds, this is true without the assumption that <span class="texhtml mvar" style="font-style:italic;">V</span> has finite dimension. This also indicates that a basis is a minimal spanning set when <span class="texhtml mvar" style="font-style:italic;">V</span> is finite-dimensional. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_span&action=edit&section=7" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Generalizing the definition of the span of points in space, a subset <span class="texhtml mvar" style="font-style:italic;">X</span> of the ground set of a <a href="/wiki/Matroid" title="Matroid">matroid</a> is called a spanning set if the rank of <span class="texhtml mvar" style="font-style:italic;">X</span> equals the rank of the entire ground set<sup id="cite_ref-FOOTNOTEOxley201128_8-0" class="reference"><a href="#cite_note-FOOTNOTEOxley201128-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The vector space definition can also be generalized to modules.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Given an <span class="texhtml mvar" style="font-style:italic;">R</span>-module <span class="texhtml mvar" style="font-style:italic;">A</span> and a collection of elements <span class="texhtml"><i>a</i><sub>1</sub></span>, ..., <span class="texhtml"><i>a<sub>n</sub></i></span> of <span class="texhtml mvar" style="font-style:italic;">A</span>, the <a href="/wiki/Submodule" class="mw-redirect" title="Submodule">submodule</a> of <span class="texhtml mvar" style="font-style:italic;">A</span> spanned by <span class="texhtml"><i>a</i><sub>1</sub></span>, ..., <span class="texhtml"><i>a<sub>n</sub></i></span> is the sum of <a href="/wiki/Cyclic_module" title="Cyclic module">cyclic modules</a> <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 Ra_{1}+\cdots +Ra_{n}=\left\{\sum _{k=1}^{n}r_{k}a_{k}{\bigg |}r_{k}\in R\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>R</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mi>R</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ra_{1}+\cdots +Ra_{n}=\left\{\sum _{k=1}^{n}r_{k}a_{k}{\bigg |}r_{k}\in R\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64242dc39bd9b91de6439a8c20dfdc3d7b747fd7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.096ex; height:7.509ex;" alt="{\displaystyle Ra_{1}+\cdots +Ra_{n}=\left\{\sum _{k=1}^{n}r_{k}a_{k}{\bigg |}r_{k}\in R\right\}}"></span> consisting of all <i>R</i>-linear combinations of the elements <span class="texhtml"><i>a<sub>i</sub></i></span>. As with the case of vector spaces, the submodule of <i>A</i> spanned by any subset of <i>A</i> is the intersection of all submodules containing that subset. </p> <div class="mw-heading mw-heading2"><h2 id="Closed_linear_span_(functional_analysis)"><span id="Closed_linear_span_.28functional_analysis.29"></span>Closed linear span (functional analysis)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_span&action=edit&section=8" title="Edit section: Closed linear span (functional analysis)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>, a closed linear span of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <a href="/wiki/Vector_space" title="Vector space">vectors</a> is the minimal closed set which contains the linear span of that set. </p><p>Suppose that <span class="texhtml mvar" style="font-style:italic;">X</span> is a normed vector space and let <span class="texhtml mvar" style="font-style:italic;">E</span> be any non-empty subset of <span class="texhtml mvar" style="font-style:italic;">X</span>. The <b>closed linear span</b> of <span class="texhtml mvar" style="font-style:italic;">E</span>, denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\operatorname {Sp} }}(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Sp</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\operatorname {Sp} }}(E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/751ef8a31871b2de0b7a3e29a68ad2a9f4f3280d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.285ex; height:3.509ex;" alt="{\displaystyle {\overline {\operatorname {Sp} }}(E)}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\operatorname {Span} }}(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Span</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\operatorname {Span} }}(E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/773d2a7eac545cd1fc0efc199a705713db4e224c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.74ex; height:3.509ex;" alt="{\displaystyle {\overline {\operatorname {Span} }}(E)}"></span>, is the intersection of all the closed linear subspaces of <span class="texhtml mvar" style="font-style:italic;">X</span> which contain <span class="texhtml mvar" style="font-style:italic;">E</span>. </p><p>One mathematical formulation of this is </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 {\overline {\operatorname {Sp} }}(E)=\{u\in X|\forall \varepsilon >0\,\exists x\in \operatorname {Sp} (E):\|x-u\|<\varepsilon \}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Sp</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>u</mi> <mo>∈</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">∀</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo>∈</mo> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>u</mi> <mo fence="false" stretchy="false">‖</mo> <mo><</mo> <mi>ε</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\operatorname {Sp} }}(E)=\{u\in X|\forall \varepsilon >0\,\exists x\in \operatorname {Sp} (E):\|x-u\|<\varepsilon \}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/395b2b3582d596bde6bd435c2a1e5e7e13691d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.753ex; height:3.509ex;" alt="{\displaystyle {\overline {\operatorname {Sp} }}(E)=\{u\in X|\forall \varepsilon >0\,\exists x\in \operatorname {Sp} (E):\|x-u\|<\varepsilon \}.}"></span></dd></dl> <p>The closed linear span of the set of functions <i>x<sup>n</sup></i> on the interval [0, 1], where <i>n</i> is a non-negative integer, depends on the norm used. If the <a href="/wiki/Lp_space#Lp_spaces_and_Lebesgue_integrals" title="Lp space"><i>L</i><sup>2</sup> norm</a> is used, then the closed linear span is the <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> of <a href="/wiki/Square-integrable_function" title="Square-integrable function">square-integrable functions</a> on the interval. But if the <a href="/wiki/Maximum_norm" class="mw-redirect" title="Maximum norm">maximum norm</a> is used, the closed linear span will be the space of continuous functions on the interval. In either case, the closed linear span contains functions that are not polynomials, and so are not in the linear span itself. However, the <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> of the set of functions in the closed linear span is the <a href="/wiki/Cardinality_of_the_continuum" title="Cardinality of the continuum">cardinality of the continuum</a>, which is the same cardinality as for the set of polynomials. </p> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_span&action=edit&section=9" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The linear span of a set is dense in the closed linear span. Moreover, as stated in the lemma below, the closed linear span is indeed the <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closure</a> of the linear span. </p><p>Closed linear spans are important when dealing with closed linear subspaces (which are themselves highly important, see <a href="/wiki/Riesz%27s_lemma" title="Riesz's lemma">Riesz's lemma</a>). </p> <div class="mw-heading mw-heading3"><h3 id="A_useful_lemma">A useful lemma</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_span&action=edit&section=10" title="Edit section: A useful lemma"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml mvar" style="font-style:italic;">X</span> be a normed space and let <span class="texhtml mvar" style="font-style:italic;">E</span> be any non-empty subset of <span class="texhtml mvar" style="font-style:italic;">X</span>. Then </p> <div><ol style="list-style-type:lower-alpha"><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\operatorname {Sp} }}(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Sp</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\operatorname {Sp} }}(E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/751ef8a31871b2de0b7a3e29a68ad2a9f4f3280d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.285ex; height:3.509ex;" alt="{\displaystyle {\overline {\operatorname {Sp} }}(E)}"></span> is a closed linear subspace of <i>X</i> which contains <i>E</i>,</li><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\operatorname {Sp} }}(E)={\overline {\operatorname {Sp} (E)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Sp</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\operatorname {Sp} }}(E)={\overline {\operatorname {Sp} (E)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f36a9ffa82425c6059cb7dbd3c6fb1da66a418c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.668ex; height:3.676ex;" alt="{\displaystyle {\overline {\operatorname {Sp} }}(E)={\overline {\operatorname {Sp} (E)}}}"></span>, viz. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\operatorname {Sp} }}(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Sp</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\operatorname {Sp} }}(E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/751ef8a31871b2de0b7a3e29a68ad2a9f4f3280d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.285ex; height:3.509ex;" alt="{\displaystyle {\overline {\operatorname {Sp} }}(E)}"></span> is the closure of <span class="mwe-math-element"><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 {Sp} (E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed11e6e9dd926fe0a7d0e98f1024a16ec70000f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.17ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sp} (E)}"></span>,</li><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{\perp }=(\operatorname {Sp} (E))^{\perp }=\left({\overline {\operatorname {Sp} (E)}}\right)^{\perp }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{\perp }=(\operatorname {Sp} (E))^{\perp }=\left({\overline {\operatorname {Sp} (E)}}\right)^{\perp }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83dc8a0b0d4febe16ee3a6d41d621308a1835883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.21ex; height:5.176ex;" alt="{\displaystyle E^{\perp }=(\operatorname {Sp} (E))^{\perp }=\left({\overline {\operatorname {Sp} (E)}}\right)^{\perp }.}"></span></li></ol></div> <p>(So the usual way to find the closed linear span is to find the linear span first, and then the closure of that linear span.) </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_span&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Affine_hull" title="Affine hull">Affine hull</a></li> <li><a href="/wiki/Conical_combination" title="Conical combination">Conical combination</a></li> <li><a href="/wiki/Convex_hull" title="Convex hull">Convex hull</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_span&action=edit&section=12" title="Edit section: Footnotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"> This is logically valid as when <span class="texhtml"><i>n</i> = 0</span>, the conditions for the vectors and constants are empty, and therefore <a href="/wiki/Vacuously" class="mw-redirect" title="Vacuously">vacuously</a> satisfied.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_span&action=edit&section=13" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><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"><a href="#CITEREFEncyclopedia_of_Mathematics2020">Encyclopedia of Mathematics (2020)</a>. Linear Hull.</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"><a href="#CITEREFAxler2015">Axler (2015)</a> p. 29, § 2.7</span> </li> <li id="cite_note-:0-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-:0_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFAxler2015">Axler (2015)</a> pp. 29-30, §§ 2.5, 2.8</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"><a href="#CITEREFHefferon2020">Hefferon (2020)</a> p. 100, ch. 2, Definition 2.13</span> </li> <li id="cite_note-:02-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-:02_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFAxler2015">Axler (2015)</a> pp. 29-30, §§ 2.5, 2.8</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFRoman2005">Roman (2005)</a> pp. 41-42</span> </li> <li id="cite_note-FOOTNOTEOxley201128-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEOxley201128_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFOxley2011">Oxley (2011)</a>, p. 28.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="#CITEREFRoman2005">Roman (2005)</a> p. 96, ch. 4</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"><a href="#CITEREFMac_LaneBirkhoff1999">Mac Lane & Birkhoff (1999)</a> p. 193, ch. 6</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_span&action=edit&section=14" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Textbooks">Textbooks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_span&action=edit&section=15" title="Edit section: Textbooks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAxler2015" class="citation book cs1"><a href="/wiki/Sheldon_Axler" title="Sheldon Axler">Axler, Sheldon Jay</a> (2015). <a rel="nofollow" class="external text" href="https://linear.axler.net/LADR4e.pdf#page=43"><i>Linear Algebra Done Right</i></a> <span class="cs1-format">(PDF)</span> (3rd ed.). <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media"> Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-11079-0" title="Special:BookSources/978-3-319-11079-0"><bdi>978-3-319-11079-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+Done+Right&rft.edition=3rd&rft.pub=Springer&rft.date=2015&rft.isbn=978-3-319-11079-0&rft.aulast=Axler&rft.aufirst=Sheldon+Jay&rft_id=https%3A%2F%2Flinear.axler.net%2FLADR4e.pdf%23page%3D43&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+span" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHefferon2020" class="citation book cs1"><a href="/wiki/Jim_Hefferon" title="Jim Hefferon">Hefferon, Jim</a> (2020). <a rel="nofollow" class="external text" href="https://www.cs.ox.ac.uk/files/12921/book.pdf#page=110"><i>Linear Algebra</i></a> <span class="cs1-format">(PDF)</span> (4th ed.). Orthogonal Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-944325-11-4" title="Special:BookSources/978-1-944325-11-4"><bdi>978-1-944325-11-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra&rft.edition=4th&rft.pub=Orthogonal+Publishing&rft.date=2020&rft.isbn=978-1-944325-11-4&rft.aulast=Hefferon&rft.aufirst=Jim&rft_id=https%3A%2F%2Fwww.cs.ox.ac.uk%2Ffiles%2F12921%2Fbook.pdf%23page%3D110&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+span" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMac_LaneBirkhoff1999" class="citation book cs1"><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Mac Lane, Saunders</a>; <a href="/wiki/Garrett_Birkhoff" title="Garrett Birkhoff">Birkhoff, Garrett</a> (1999) [1988]. <i>Algebra</i> (3rd ed.). <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">AMS Chelsea Publishing</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0821816462" title="Special:BookSources/978-0821816462"><bdi>978-0821816462</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.edition=3rd&rft.pub=AMS+Chelsea+Publishing&rft.date=1999&rft.isbn=978-0821816462&rft.aulast=Mac+Lane&rft.aufirst=Saunders&rft.au=Birkhoff%2C+Garrett&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+span" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOxley2011" class="citation book cs1"><a href="/wiki/James_Oxley" title="James Oxley">Oxley, James G.</a> (2011). <i>Matroid Theory</i>. Oxford Graduate Texts in Mathematics. Vol. 3 (2nd ed.). Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780199202508" title="Special:BookSources/9780199202508"><bdi>9780199202508</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matroid+Theory&rft.series=Oxford+Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Oxford+University+Press&rft.date=2011&rft.isbn=9780199202508&rft.aulast=Oxley&rft.aufirst=James+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+span" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoman2005" class="citation book cs1"><a href="/wiki/Steven_Roman" title="Steven Roman">Roman, Steven</a> (2005). <a rel="nofollow" class="external text" href="http://matematicas.uis.edu.co/sites/default/files/paginas/archivos/Advanced%20Linear%20Algebra%20-%20Steven%20Roman.pdf#page=56"><i>Advanced Linear Algebra</i></a> <span class="cs1-format">(PDF)</span> (2nd ed.). <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-24766-1" title="Special:BookSources/0-387-24766-1"><bdi>0-387-24766-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Linear+Algebra&rft.edition=2nd&rft.pub=Springer&rft.date=2005&rft.isbn=0-387-24766-1&rft.aulast=Roman&rft.aufirst=Steven&rft_id=http%3A%2F%2Fmatematicas.uis.edu.co%2Fsites%2Fdefault%2Ffiles%2Fpaginas%2Farchivos%2FAdvanced%2520Linear%2520Algebra%2520-%2520Steven%2520Roman.pdf%23page%3D56&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+span" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRynneYoungson2008" class="citation book cs1">Rynne, Brian P.; Youngson, Martin A. (2008). <i>Linear Functional Analysis</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1848000049" title="Special:BookSources/978-1848000049"><bdi>978-1848000049</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Functional+Analysis&rft.pub=Springer&rft.date=2008&rft.isbn=978-1848000049&rft.aulast=Rynne&rft.aufirst=Brian+P.&rft.au=Youngson%2C+Martin+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+span" class="Z3988"></span></li> <li>Lay, David C. (2021) <i>Linear Algebra and Its Applications (6th Edition)</i>. Pearson.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Web">Web</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_span&action=edit&section=16" title="Edit section: Web"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLankhamNachtergaeleSchilling2010" class="citation web cs1">Lankham, Isaiah; <a href="/wiki/Bruno_Nachtergaele" title="Bruno Nachtergaele">Nachtergaele, Bruno</a>; <a href="/wiki/Anne_Schilling" title="Anne Schilling">Schilling, Anne</a> (13 February 2010). <a rel="nofollow" class="external text" href="https://www.math.ucdavis.edu/~anne/linear_algebra/mat67_course_notes.pdf">"Linear Algebra - As an Introduction to Abstract Mathematics"</a> <span class="cs1-format">(PDF)</span>. University of California, Davis<span class="reference-accessdate">. Retrieved <span class="nowrap">27 September</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Linear+Algebra+-+As+an+Introduction+to+Abstract+Mathematics&rft.pub=University+of+California%2C+Davis&rft.date=2010-02-13&rft.aulast=Lankham&rft.aufirst=Isaiah&rft.au=Nachtergaele%2C+Bruno&rft.au=Schilling%2C+Anne&rft_id=https%3A%2F%2Fwww.math.ucdavis.edu%2F~anne%2Flinear_algebra%2Fmat67_course_notes.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+span" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMathWorld2021" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric Wolfgang</a>. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/VectorSpaceSpan.html">"Vector Space Span"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">16 Feb</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Vector+Space+Span&rft.aulast=Weisstein&rft.aufirst=Eric+Wolfgang&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FVectorSpaceSpan.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+span" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEncyclopedia_of_Mathematics2020" class="citation web cs1"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Linear_hull">"Linear hull"</a>. <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>. 5 April 2020<span class="reference-accessdate">. Retrieved <span class="nowrap">16 Feb</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Linear+hull&rft.date=2020-04-05&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FLinear_hull&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+span" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_span&action=edit&section=17" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/linear_combinations/v/linear-combinations-and-span">Linear Combinations and Span: Understanding linear combinations and spans of vectors</a>, khanacademy.org.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSanderson2016" class="citation web cs1"><a href="/wiki/3Blue1Brown" title="3Blue1Brown">Sanderson, Grant</a> (August 6, 2016). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=k7RM-ot2NWY&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=3">"Linear combinations, span, and basis vectors"</a>. Essence of Linear Algebra. <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211211/k7RM-ot2NWY">Archived</a> from the original on 2021-12-11 – via <a href="/wiki/YouTube" title="YouTube">YouTube</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Linear+combinations%2C+span%2C+and+basis+vectors&rft.series=Essence+of+Linear+Algebra&rft.date=2016-08-06&rft.aulast=Sanderson&rft.aufirst=Grant&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3Dk7RM-ot2NWY%26list%3DPLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab%26index%3D3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+span" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.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 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 dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.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 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.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{padding:0.25em 1em;line-height:1.5em;text-align:center}.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{line-height:1.5em;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}.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}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="Linear_algebra" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;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:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}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:Linear_algebra" title="Template:Linear algebra"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Linear_algebra" title="Template talk:Linear algebra"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Linear_algebra" title="Special:EditPage/Template:Linear algebra"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Linear_algebra" style="font-size:114%;margin:0 4em"><a href="/wiki/Linear_algebra" title="Linear algebra">Linear algebra</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><a href="/wiki/Outline_of_linear_algebra" title="Outline of linear algebra">Outline</a></li> <li><a href="/wiki/Glossary_of_linear_algebra" title="Glossary of linear algebra">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">Scalar</a></li> <li><a href="/wiki/Euclidean_vector" title="Euclidean vector">Vector</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li> <li><a href="/wiki/Scalar_multiplication" title="Scalar multiplication">Scalar multiplication</a></li> <li><a href="/wiki/Vector_projection" title="Vector projection">Vector projection</a></li> <li><a class="mw-selflink selflink">Linear span</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">Linear projection</a></li> <li><a href="/wiki/Linear_independence" title="Linear independence">Linear independence</a></li> <li><a href="/wiki/Linear_combination" title="Linear combination">Linear combination</a></li> <li><a href="/wiki/Multilinear_map" title="Multilinear map">Multilinear map</a></li> <li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Change_of_basis" title="Change of basis">Change of basis</a></li> <li><a href="/wiki/Row_and_column_vectors" title="Row and column vectors">Row and column vectors</a></li> <li><a href="/wiki/Row_and_column_spaces" title="Row and column spaces">Row and column spaces</a></li> <li><a href="/wiki/Kernel_(linear_algebra)" title="Kernel (linear algebra)">Kernel</a></li> <li><a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">Eigenvalues and eigenvectors</a></li> <li><a href="/wiki/Transpose" title="Transpose">Transpose</a></li> <li><a href="/wiki/System_of_linear_equations" title="System of linear equations">Linear equations</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Euclidean_space" title="Euclidean space"><img alt="Three dimensional Euclidean space" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/80px-Linear_subspaces_with_shading.svg.png" decoding="async" width="80" height="58" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/120px-Linear_subspaces_with_shading.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/160px-Linear_subspaces_with_shading.svg.png 2x" data-file-width="325" data-file-height="236" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrices</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Block_matrix" title="Block matrix">Block</a></li> <li><a href="/wiki/Matrix_decomposition" title="Matrix decomposition">Decomposition</a></li> <li><a href="/wiki/Invertible_matrix" title="Invertible matrix">Invertible</a></li> <li><a href="/wiki/Minor_(linear_algebra)" title="Minor (linear algebra)">Minor</a></li> <li><a href="/wiki/Matrix_multiplication" title="Matrix multiplication">Multiplication</a></li> <li><a href="/wiki/Rank_(linear_algebra)" title="Rank (linear algebra)">Rank</a></li> <li><a href="/wiki/Transformation_matrix" title="Transformation matrix">Transformation</a></li> <li><a href="/wiki/Cramer%27s_rule" title="Cramer's rule">Cramer's rule</a></li> <li><a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a></li> <li><a href="/wiki/Productive_matrix" title="Productive matrix">Productive matrix</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Bilinear_map" title="Bilinear map">Bilinear</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Orthogonality" title="Orthogonality">Orthogonality</a></li> <li><a href="/wiki/Dot_product" title="Dot product">Dot product</a></li> <li><a href="/wiki/Hadamard_product_(matrices)" title="Hadamard product (matrices)">Hadamard product</a></li> <li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a></li> <li><a href="/wiki/Outer_product" title="Outer product">Outer product</a></li> <li><a href="/wiki/Kronecker_product" title="Kronecker product">Kronecker product</a></li> <li><a href="/wiki/Gram%E2%80%93Schmidt_process" title="Gram–Schmidt process">Gram–Schmidt process</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Determinant" title="Determinant">Determinant</a></li> <li><a href="/wiki/Cross_product" title="Cross product">Cross product</a></li> <li><a href="/wiki/Triple_product" title="Triple product">Triple product</a></li> <li><a href="/wiki/Seven-dimensional_cross_product" title="Seven-dimensional cross product">Seven-dimensional cross product</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a></li> <li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior algebra</a></li> <li><a href="/wiki/Bivector" title="Bivector">Bivector</a></li> <li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li> <li><a href="/wiki/Outermorphism" title="Outermorphism">Outermorphism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_space" title="Vector space">Vector space</a> constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dual_space" title="Dual space">Dual</a></li> <li><a href="/wiki/Direct_sum_of_modules#Construction_for_two_vector_spaces" title="Direct sum of modules">Direct sum</a></li> <li><a href="/wiki/Function_space#In_linear_algebra" title="Function space">Function space</a></li> <li><a href="/wiki/Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">Quotient</a></li> <li><a href="/wiki/Linear_subspace" title="Linear subspace">Subspace</a></li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">Numerical</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">Floating-point</a></li> <li><a href="/wiki/Numerical_stability" title="Numerical stability">Numerical stability</a></li> <li><a href="/wiki/Basic_Linear_Algebra_Subprograms" title="Basic Linear Algebra Subprograms">Basic Linear Algebra Subprograms</a></li> <li><a href="/wiki/Sparse_matrix" title="Sparse matrix">Sparse matrix</a></li> <li><a href="/wiki/Comparison_of_linear_algebra_libraries" title="Comparison of linear algebra libraries">Comparison of linear algebra libraries</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Linear_algebra" title="Category:Linear algebra">Category</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="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Linear_span&oldid=1251730161">https://en.wikipedia.org/w/index.php?title=Linear_span&oldid=1251730161</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Abstract_algebra" title="Category:Abstract algebra">Abstract algebra</a></li><li><a href="/wiki/Category:Linear_algebra" title="Category:Linear algebra">Linear algebra</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</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"> This page was last edited on 17 October 2024, at 19:15<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Linear_span&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</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-f69cdc8f6-pnlkl","wgBackendResponseTime":127,"wgPageParseReport":{"limitreport":{"cputime":"0.449","walltime":"0.667","ppvisitednodes":{"value":3215,"limit":1000000},"postexpandincludesize":{"value":43130,"limit":2097152},"templateargumentsize":{"value":2722,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":1,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":37510,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 424.432 1 -total"," 29.39% 124.745 6 Template:Cite_book"," 20.96% 88.974 1 Template:Linear_algebra"," 20.54% 87.188 1 Template:Navbox"," 16.37% 69.478 1 Template:Short_description"," 10.58% 44.885 2 Template:Pagetype"," 7.58% 32.151 1 Template:Sfnp"," 5.69% 24.141 23 Template:Math"," 4.58% 19.424 4 Template:Cite_web"," 4.16% 17.635 28 Template:Main_other"]},"scribunto":{"limitreport-timeusage":{"value":"0.229","limit":"10.000"},"limitreport-memusage":{"value":6542685,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAxler2015\"] = 1,\n [\"CITEREFEncyclopedia_of_Mathematics2020\"] = 1,\n [\"CITEREFHefferon2020\"] = 1,\n [\"CITEREFLankhamNachtergaeleSchilling2010\"] = 1,\n [\"CITEREFMac_LaneBirkhoff1999\"] = 1,\n [\"CITEREFMathWorld2021\"] = 1,\n [\"CITEREFOxley2011\"] = 1,\n [\"CITEREFRoman2005\"] = 1,\n [\"CITEREFRynneYoungson2008\"] = 1,\n [\"CITEREFSanderson2016\"] = 1,\n}\ntemplate_list = table#1 {\n [\"Cbignore\"] = 1,\n [\"Cite book\"] = 6,\n [\"Cite web\"] = 4,\n [\"Efn\"] = 1,\n [\"Frac\"] = 1,\n [\"Harvard citation text\"] = 8,\n [\"Linear algebra\"] = 1,\n [\"Math\"] = 23,\n [\"Mvar\"] = 92,\n [\"Notelist\"] = 1,\n [\"Ordered list\"] = 1,\n [\"Sfnp\"] = 1,\n [\"Short description\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-9s56p","timestamp":"20241124161611","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Linear span","url":"https:\/\/en.wikipedia.org\/wiki\/Linear_span","sameAs":"http:\/\/www.wikidata.org\/entity\/Q209812","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q209812","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-06-30T20:20:11Z","dateModified":"2024-10-17T19:15:17Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/7\/70\/Basis_for_a_plane.svg","headline":"smallest vector subspace containing a given subspace"}</script> </body> </html>