Affine space - Wikipedia

<!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>Affine space - 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":"73ecc5a9-1ca4-4810-8138-2df0e23fc2ba","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Affine_space","wgTitle":"Affine space","wgCurRevisionId":1251420333,"wgRevisionId":1251420333,"wgArticleId":298834,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["CS1 maint: date and year","Articles with short description","Short description is different from Wikidata","Affine geometry","Linear algebra","Space (mathematics)"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Affine_space","wgRelevantArticleId":298834,"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":50000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q382698","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.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"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","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.quicksurveys.init","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%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/9/95/Affine_space_R3.png/1200px-Affine_space_R3.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="830"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/9/95/Affine_space_R3.png/800px-Affine_space_R3.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="553"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/9/95/Affine_space_R3.png/640px-Affine_space_R3.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="443"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Affine space - 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/Affine_space"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Affine_space&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/Affine_space"> <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-Affine_space rootpage-Affine_space 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" > Main menu </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">Main page</a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia">Contents</a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events">Current events</a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x">Random article</a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works">About Wikipedia</a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia">Contact us</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">Help</a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia">Learn to edit</a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors">Community portal</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">Recent changes</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">Upload file</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"> <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;"> </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"> Search </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" > </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" > Appearance </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="">Donate</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=Affine+space" title="You are encouraged to create an account and log in; however, it is not mandatory" class="">Create account</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=Affine+space" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class="">Log in</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" > Personal tools </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">Donate</a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Affine+space" title="You are encouraged to create an account and log in; however, it is not mandatory"> Create account</a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Affine+space" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"> Log in</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">learn more</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">Contributions</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">Talk</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-Informal_description" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Informal_description"> <div class="vector-toc-text"> 1 Informal description </div> </a> <ul id="toc-Informal_description-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> 2 Definition </div> </a> <button aria-controls="toc-Definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Definition subsection </button> <ul id="toc-Definition-sublist" class="vector-toc-list"> <li id="toc-Subtraction_and_Weyl's_axioms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subtraction_and_Weyl's_axioms"> <div class="vector-toc-text"> 2.1 Subtraction and Weyl's axioms </div> </a> <ul id="toc-Subtraction_and_Weyl's_axioms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Affine_subspaces_and_parallelism" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Affine_subspaces_and_parallelism"> <div class="vector-toc-text"> 3 Affine subspaces and parallelism </div> </a> <ul id="toc-Affine_subspaces_and_parallelism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Affine_map" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Affine_map"> <div class="vector-toc-text"> 4 Affine map </div> </a> <button aria-controls="toc-Affine_map-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Affine map subsection </button> <ul id="toc-Affine_map-sublist" class="vector-toc-list"> <li id="toc-Endomorphisms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Endomorphisms"> <div class="vector-toc-text"> 4.1 Endomorphisms </div> </a> <ul id="toc-Endomorphisms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vector_spaces_as_affine_spaces" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Vector_spaces_as_affine_spaces"> <div class="vector-toc-text"> 5 Vector spaces as affine spaces </div> </a> <ul id="toc-Vector_spaces_as_affine_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_Euclidean_spaces" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relation_to_Euclidean_spaces"> <div class="vector-toc-text"> 6 Relation to Euclidean spaces </div> </a> <button aria-controls="toc-Relation_to_Euclidean_spaces-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Relation to Euclidean spaces subsection </button> <ul id="toc-Relation_to_Euclidean_spaces-sublist" class="vector-toc-list"> <li id="toc-Definition_of_Euclidean_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition_of_Euclidean_spaces"> <div class="vector-toc-text"> 6.1 Definition of Euclidean spaces </div> </a> <ul id="toc-Definition_of_Euclidean_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Affine_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Affine_properties"> <div class="vector-toc-text"> 6.2 Affine properties </div> </a> <ul id="toc-Affine_properties-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Affine_combinations_and_barycenter" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Affine_combinations_and_barycenter"> <div class="vector-toc-text"> 7 Affine combinations and barycenter </div> </a> <ul id="toc-Affine_combinations_and_barycenter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> 8 Examples </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Affine_span_and_bases" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Affine_span_and_bases"> <div class="vector-toc-text"> 9 Affine span and bases </div> </a> <ul id="toc-Affine_span_and_bases-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coordinates" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Coordinates"> <div class="vector-toc-text"> 10 Coordinates </div> </a> <button aria-controls="toc-Coordinates-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Coordinates subsection </button> <ul id="toc-Coordinates-sublist" class="vector-toc-list"> <li id="toc-Barycentric_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Barycentric_coordinates"> <div class="vector-toc-text"> 10.1 Barycentric coordinates </div> </a> <ul id="toc-Barycentric_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Affine_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Affine_coordinates"> <div class="vector-toc-text"> 10.2 Affine coordinates </div> </a> <ul id="toc-Affine_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_between_barycentric_and_affine_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relationship_between_barycentric_and_affine_coordinates"> <div class="vector-toc-text"> 10.3 Relationship between barycentric and affine coordinates </div> </a> <ul id="toc-Relationship_between_barycentric_and_affine_coordinates-sublist" class="vector-toc-list"> <li id="toc-Example_of_the_triangle" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_of_the_triangle"> <div class="vector-toc-text"> 10.3.1 Example of the triangle </div> </a> <ul id="toc-Example_of_the_triangle-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Change_of_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Change_of_coordinates"> <div class="vector-toc-text"> 10.4 Change of coordinates </div> </a> <ul id="toc-Change_of_coordinates-sublist" class="vector-toc-list"> <li id="toc-Case_of_barycentric_coordinates" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Case_of_barycentric_coordinates"> <div class="vector-toc-text"> 10.4.1 Case of barycentric coordinates </div> </a> <ul id="toc-Case_of_barycentric_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Case_of_affine_coordinates" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Case_of_affine_coordinates"> <div class="vector-toc-text"> 10.4.2 Case of affine coordinates </div> </a> <ul id="toc-Case_of_affine_coordinates-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Properties_of_affine_homomorphisms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties_of_affine_homomorphisms"> <div class="vector-toc-text"> 11 Properties of affine homomorphisms </div> </a> <button aria-controls="toc-Properties_of_affine_homomorphisms-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties of affine homomorphisms subsection </button> <ul id="toc-Properties_of_affine_homomorphisms-sublist" class="vector-toc-list"> <li id="toc-Matrix_representation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matrix_representation"> <div class="vector-toc-text"> 11.1 Matrix representation </div> </a> <ul id="toc-Matrix_representation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Image_and_fibers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Image_and_fibers"> <div class="vector-toc-text"> 11.2 Image and fibers </div> </a> <ul id="toc-Image_and_fibers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Projection" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Projection"> <div class="vector-toc-text"> 11.3 Projection </div> </a> <ul id="toc-Projection-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quotient_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quotient_space"> <div class="vector-toc-text"> 11.4 Quotient space </div> </a> <ul id="toc-Quotient_space-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Axioms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Axioms"> <div class="vector-toc-text"> 12 Axioms </div> </a> <ul id="toc-Axioms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_projective_spaces" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relation_to_projective_spaces"> <div class="vector-toc-text"> 13 Relation to projective spaces </div> </a> <ul id="toc-Relation_to_projective_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Affine_algebraic_geometry" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Affine_algebraic_geometry"> <div class="vector-toc-text"> 14 Affine algebraic geometry </div> </a> <button aria-controls="toc-Affine_algebraic_geometry-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Affine algebraic geometry subsection </button> <ul id="toc-Affine_algebraic_geometry-sublist" class="vector-toc-list"> <li id="toc-Ring_of_polynomial_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ring_of_polynomial_functions"> <div class="vector-toc-text"> 14.1 Ring of polynomial functions </div> </a> <ul id="toc-Ring_of_polynomial_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zariski_topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zariski_topology"> <div class="vector-toc-text"> 14.2 Zariski topology </div> </a> <ul id="toc-Zariski_topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cohomology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cohomology"> <div class="vector-toc-text"> 14.3 Cohomology </div> </a> <ul id="toc-Cohomology-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 15 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 16 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 17 References </div> </a> <ul id="toc-References-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" > Toggle the table of contents </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">Affine space</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 28 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-28" aria-hidden="true" > 28 languages </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%81%D8%B6%D8%A7%D8%A1_%D8%AA%D8%A2%D9%84%D9%81%D9%8A" title="فضاء تآلفي – Arabic" lang="ar" hreflang="ar" data-title="فضاء تآلفي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D1%84%D0%B8%D0%BD%D0%BD%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Афинно пространство – Bulgarian" lang="bg" hreflang="bg" data-title="Афинно пространство" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_af%C3%AD" title="Espai afí – Catalan" lang="ca" hreflang="ca" data-title="Espai afí" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%90%D1%84%D1%84%D0%B8%D0%BD%D0%BB%D0%B0_%D1%83%C3%A7%D0%BB%C4%83%D1%85" title="Аффинла уçлăх – Chuvash" lang="cv" hreflang="cv" data-title="Аффинла уçлăх" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Afinn%C3%AD_prostor" title="Afinní prostor – Czech" lang="cs" hreflang="cs" data-title="Afinní prostor" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gofod_affin" title="Gofod affin – Welsh" lang="cy" hreflang="cy" data-title="Gofod affin" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Affiner_Raum" title="Affiner Raum – German" lang="de" hreflang="de" data-title="Affiner Raum" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Afiinne_ruum" title="Afiinne ruum – Estonian" lang="et" hreflang="et" data-title="Afiinne ruum" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%86%CE%B9%CE%BD%CE%B9%CE%BA%CF%8C%CF%82_%CF%87%CF%8E%CF%81%CE%BF%CF%82" title="Αφινικός χώρος – Greek" lang="el" hreflang="el" data-title="Αφινικός χώρος" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espacio_af%C3%ADn" title="Espacio afín – Spanish" lang="es" hreflang="es" data-title="Espacio afín" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%DB%8C_%D8%A2%D9%81%DB%8C%D9%86" title="فضای آفین – Persian" lang="fa" hreflang="fa" data-title="فضای آفین" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Espace_affine" title="Espace affine – French" lang="fr" hreflang="fr" data-title="Espace affine" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%95%84%ED%95%80_%EA%B3%B5%EA%B0%84" title="아핀 공간 – Korean" lang="ko" hreflang="ko" data-title="아핀 공간" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_affine" title="Spazio affine – Italian" lang="it" hreflang="it" data-title="Spazio affine" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%90%D7%A4%D7%99%D7%A0%D7%99" title="מרחב אפיני – Hebrew" lang="he" hreflang="he" data-title="מרחב אפיני" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%90%D1%84%D0%B8%D0%BD_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Афин простор – Macedonian" lang="mk" hreflang="mk" data-title="Афин простор" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Affiene_ruimte" title="Affiene ruimte – Dutch" lang="nl" hreflang="nl" data-title="Affiene ruimte" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A2%E3%83%95%E3%82%A3%E3%83%B3%E7%A9%BA%E9%96%93" title="アフィン空間 – Japanese" lang="ja" hreflang="ja" data-title="アフィン空間" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Affint_rom" title="Affint rom – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Affint rom" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%85%E0%A9%B1%E0%A8%AB%E0%A8%BE%E0%A8%88%E0%A8%A8_%E0%A8%B8%E0%A8%AA%E0%A9%87%E0%A8%B8" title="ਅੱਫਾਈਨ ਸਪੇਸ – Punjabi" lang="pa" hreflang="pa" data-title="ਅੱਫਾਈਨ ਸਪੇਸ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przestrze%C5%84_afiniczna" title="Przestrzeń afiniczna – Polish" lang="pl" hreflang="pl" data-title="Przestrzeń afiniczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_afim" title="Espaço afim – Portuguese" lang="pt" hreflang="pt" data-title="Espaço afim" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Spa%C8%9Biu_afin" title="Spațiu afin – Romanian" lang="ro" hreflang="ro" data-title="Spațiu afin" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D1%84%D1%84%D0%B8%D0%BD%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Аффинное пространство – Russian" lang="ru" hreflang="ru" data-title="Аффинное пространство" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Afinn%C3%BD_priestor" title="Afinný priestor – Slovak" lang="sk" hreflang="sk" data-title="Afinný priestor" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D1%84%D1%96%D0%BD%D0%BD%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Афінний простір – Ukrainian" lang="uk" hreflang="uk" data-title="Афінний простір" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kh%C3%B4ng_gian_afin" title="Không gian afin – Vietnamese" lang="vi" hreflang="vi" data-title="Không gian afin" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BB%BF%E5%B0%84%E7%A9%BA%E9%97%B4" title="仿射空间 – Chinese" lang="zh" hreflang="zh" data-title="仿射空间" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q382698#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></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/Affine_space" title="View the content page [c]" accesskey="c">Article</a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Affine_space" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t">Talk</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" >English </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/Affine_space">Read</a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Affine_space&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Affine_space&action=history" title="Past revisions of this page [h]" accesskey="h">View history</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" >Tools </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/Affine_space">Read</a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Affine_space&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Affine_space&action=history">View history</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/Affine_space" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j">What links here</a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Affine_space" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k">Related changes</a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u">Upload file</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">Special pages</a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Affine_space&oldid=1251420333" title="Permanent link to this revision of this page">Permanent link</a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Affine_space&action=info" title="More information about this page">Page information</a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Affine_space&id=1251420333&wpFormIdentifier=titleform" title="Information on how to cite this page">Cite this page</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%2FAffine_space">Get shortened URL</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%2FAffine_space">Download QR code</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=Affine_space&action=show-download-screen" title="Download this page as a PDF file">Download as PDF</a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Affine_space&printable=yes" title="Printable version of this page [p]" accesskey="p">Printable version</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/Q382698" title="Structured data on this page hosted by Wikidata [g]" accesskey="g">Wikidata item</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">Euclidean space without distance and angles</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Affinity_space" title="Affinity space">affinity space</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Affine_space_R3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Affine_space_R3.png/220px-Affine_space_R3.png" decoding="async" width="220" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Affine_space_R3.png/330px-Affine_space_R3.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/Affine_space_R3.png/440px-Affine_space_R3.png 2x" data-file-width="1562" data-file-height="1080" /></a><figcaption>In <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deb17c1074c77de2cf88d45bcd6d7a795b0f5d44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.379ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{3},}"> the upper plane (in blue)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87858df7457aa93caaef5a316db87a7240cc8c29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{2}}"> is not a vector subspace, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} \notin P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>∉</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} \notin P_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f64184fd648e060dddad8ca72363f6fc4caafb43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.724ex; height:2.676ex;" alt="{\displaystyle \mathbf {0} \notin P_{2}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} +\mathbf {b} \notin P_{2};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>∉</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} +\mathbf {b} \notin P_{2};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e539792d8680b622dee7936cbe7550805c7d0e8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.659ex; height:2.676ex;" alt="{\displaystyle \mathbf {a} +\mathbf {b} \notin P_{2};}"> it is an affine subspace. Its direction (the linear subspace associated with this affine subspace) is the lower (green) plane <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a8639b829aa80303b1521a943af1ee608b7f6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.193ex; height:2.509ex;" alt="{\displaystyle P_{1},}">, which is a vector subspace. Although <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }"> are in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d38d26ef607b31d8cde52681503bfae5ebf30f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.193ex; height:2.509ex;" alt="{\displaystyle P_{2},}"> their difference is a displacement vector, which does not belong to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d38d26ef607b31d8cde52681503bfae5ebf30f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.193ex; height:2.509ex;" alt="{\displaystyle P_{2},}"> but belongs to vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9310acf4e6f71225656b55944fb17d646a567b2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.193ex; height:2.509ex;" alt="{\displaystyle P_{1}.}"></figcaption></figure> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an affine space is a <a href="/wiki/Geometry" title="Geometry">geometric</a> <a href="/wiki/Structure_(mathematics)" class="mw-redirect" title="Structure (mathematics)">structure</a> that generalizes some of the properties of <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean spaces</a> in such a way that these are independent of the concepts of <a href="/wiki/Distance_(mathematics)" class="mw-redirect" title="Distance (mathematics)">distance</a> and measure of <a href="/wiki/Angle" title="Angle">angles</a>, keeping only the properties related to <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallelism</a> and <a href="/wiki/Ratio" title="Ratio">ratio</a> of lengths for parallel <a href="/wiki/Line_segment" title="Line segment">line segments</a>. Affine space is the setting for <a href="/wiki/Affine_geometry" title="Affine geometry">affine geometry</a>. As in Euclidean space, the fundamental objects in an affine space are called <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a>, which can be thought of as locations in the space without any size or shape: zero-<a href="/wiki/Dimension" title="Dimension">dimensional</a>. Through any pair of points an infinite <a href="/wiki/Straight_line" class="mw-redirect" title="Straight line">straight line</a> can be drawn, a one-dimensional set of points; through any three points that are not collinear, a two-dimensional <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a> can be drawn; and, in general, through k + 1 points in general position, a k-dimensional <a href="/wiki/Flat_(geometry)" title="Flat (geometry)">flat</a> or affine subspace can be drawn. Affine space is characterized by a notion of pairs of parallel lines that lie within the same plane but never meet each-other (non-parallel lines within the same plane <a href="/wiki/Intersection_(geometry)" title="Intersection (geometry)">intersect</a> in a point). Given any line, a line parallel to it can be drawn through any point in the space, and the <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence class</a> of parallel lines are said to share a direction. Unlike for vectors in a <a href="/wiki/Vector_space" title="Vector space">vector space</a>, in an affine space there is no distinguished point that serves as an <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a>. There is no predefined concept of adding or multiplying points together, or multiplying a point by a scalar number. However, for any affine space, an associated vector space can be constructed from the differences between start and end points, which are called <a href="/wiki/Free_vector" class="mw-redirect" title="Free vector">free vectors</a>, <a href="/wiki/Displacement_vector" class="mw-redirect" title="Displacement vector">displacement vectors</a>, <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a> vectors or simply translations.<a href="#cite_note-1">[1]</a> Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector. While points cannot be arbitrarily added together, it is meaningful to take <a href="/wiki/Affine_combination" title="Affine combination">affine combinations</a> of points: weighted sums with numerical coefficients summing to 1, resulting in another point. These coefficients define a <a href="/wiki/Barycentric_coordinate_system" title="Barycentric coordinate system">barycentric coordinate system</a> for the flat through the points. Any <a href="/wiki/Vector_space" title="Vector space">vector space</a> may be viewed as an affine space; this amounts to "forgetting" the special role played by the <a href="/wiki/Zero_vector" class="mw-redirect" title="Zero vector">zero vector</a>. In this case, elements of the vector space may be viewed either as points of the affine space or as displacement vectors or translations. When considered as a point, the zero vector is called the origin. Adding a fixed vector to the elements of a <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> (vector subspace) of a <a href="/wiki/Vector_space" title="Vector space">vector space</a> produces an affine subspace of the vector space. One commonly says that this affine subspace has been obtained by translating (away from the origin) the linear subspace by the translation vector (the vector added to all the elements of the linear space). In finite dimensions, such an affine subspace is the solution set of an <a href="/wiki/System_of_linear_equations#Homogeneous_systems" title="System of linear equations">inhomogeneous</a> linear system. The displacement vectors for that affine space are the solutions of the corresponding homogeneous linear system, which is a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space. The dimension of an affine space is defined as the <a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">dimension of the vector space</a> of its translations. An affine space of dimension one is an affine line. An affine space of dimension 2 is an <a href="/wiki/Affine_plane" title="Affine plane">affine plane</a>. An affine subspace of dimension n – 1 in an affine space or a vector space of dimension n is an <a href="/wiki/Affine_hyperplane" class="mw-redirect" title="Affine hyperplane">affine hyperplane</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Informal_description">Informal description</h2>[<a href="/w/index.php?title=Affine_space&action=edit&section=1" title="Edit section: Informal description">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Affine_origin.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Affine_origin.png/290px-Affine_origin.png" decoding="async" width="290" height="190" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/f/ff/Affine_origin.png 1.5x" data-file-width="421" data-file-height="276" /></a><figcaption>Origins from Alice's and Bob's perspectives. Vector computation from Alice's perspective is in red, whereas that from Bob's is in blue.</figcaption></figure> The following <a href="/wiki/Characterization_(mathematics)" title="Characterization (mathematics)">characterization</a> may be easier to understand than the usual formal definition: an affine space is what is left of a <a href="/wiki/Vector_space" title="Vector space">vector space</a> after one has forgotten which point is the origin (or, in the words of the French mathematician <a href="/wiki/Marcel_Berger" title="Marcel Berger">Marcel Berger</a>, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translations</a> to the linear maps"<a href="#cite_note-2">[2]</a>). Imagine that Alice knows that a certain point is the actual origin, but Bob believes that another point—call it p—is the origin. Two vectors, a and b, are to be added. Bob draws an arrow from point p to point a and another arrow from point p to point b, and completes the parallelogram to find what Bob thinks is a + b, but Alice knows that he has actually computed <dl><dd>p + (a − p) + (b − p).</dd></dl> Similarly, <a href="/wiki/Alice_and_Bob" title="Alice and Bob">Alice and Bob</a> may evaluate any <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of a and b, or of any finite set of vectors, and will generally get different answers. However, if the sum of the coefficients in a linear combination is 1, then Alice and Bob will arrive at the same answer. If Alice travels to <dl><dd>λa + (1 − λ)b</dd></dl> then Bob can similarly travel to <dl><dd>p + λ(a − p) + (1 − λ)(b − p) = λa + (1 − λ)b.</dd></dl> Under this condition, for all coefficients λ + (1 − λ) = 1, Alice and Bob describe the same point with the same linear combination, despite using different origins. While only Alice knows the "linear structure", both Alice and Bob know the "affine structure"—i.e. the values of <a href="/wiki/Affine_combination" title="Affine combination">affine combinations</a>, defined as linear combinations in which the sum of the coefficients is 1. A set with an affine structure is an affine space. <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Affine_space&action=edit&section=2" title="Edit section: Definition">edit</a>]</div> While affine space can be defined axiomatically (see <a href="#Axioms">§ Axioms</a> below), analogously to the definition of Euclidean space implied by <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a>, for convenience most modern sources define affine spaces in terms of the well developed vector space theory. An affine space is a set A together with a <a href="/wiki/Vector_space" title="Vector space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79a63d1d5ad20924a9e908d0f805f2b16e745dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.647ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {A}}}">, and a transitive and free <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">action</a> of the <a href="/wiki/Additive_group" title="Additive group">additive group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79a63d1d5ad20924a9e908d0f805f2b16e745dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.647ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {A}}}"> on the set A.<a href="#cite_note-3">[3]</a> The elements of the affine space A are called points. The vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79a63d1d5ad20924a9e908d0f805f2b16e745dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.647ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {A}}}"> is said to be associated to the affine space, and its elements are called vectors, translations, or sometimes <a href="/wiki/Free_vector" class="mw-redirect" title="Free vector">free vectors</a>. Explicitly, the definition above means that the action is a mapping, generally denoted as an addition, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}A\times {\overrightarrow {A}}&\to A\\(a,v)\;&\mapsto a+v,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>A</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mi>A</mi> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mi>a</mi> <mo>+</mo> <mi>v</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A\times {\overrightarrow {A}}&\to A\\(a,v)\;&\mapsto a+v,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02f8aed9158a31e5ca4c74e142bbf8911d4bbd6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.44ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}A\times {\overrightarrow {A}}&\to A\\(a,v)\;&\mapsto a+v,\end{aligned}}}"></dd></dl> that has the following properties.<a href="#cite_note-Berger1987-4">[4]</a><a href="#cite_note-5">[5]</a><a href="#cite_note-6">[6]</a> <ol><li><a href="/wiki/Right_identity" class="mw-redirect" title="Right identity">Right identity</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall a\in A,\;a+0=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>∈</mo> <mi>A</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>a</mi> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a\in A,\;a+0=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fb70fd09bcf3008f5296884b6788d7f61f49b2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.346ex; height:2.509ex;" alt="{\displaystyle \forall a\in A,\;a+0=a}">, where 0 is the zero vector in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79a63d1d5ad20924a9e908d0f805f2b16e745dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.647ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {A}}}"></dd></dl></li> <li><a href="/wiki/Associativity" class="mw-redirect" title="Associativity">Associativity</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall v,w\in {\overrightarrow {A}},\forall a\in A,\;(a+v)+w=a+(v+w)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> <mo>,</mo> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>∈</mo> <mi>A</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>w</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo>+</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall v,w\in {\overrightarrow {A}},\forall a\in A,\;(a+v)+w=a+(v+w)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea1f3c8ae46c9de65576544b3054b22ae5033b6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.546ex; height:4.176ex;" alt="{\displaystyle \forall v,w\in {\overrightarrow {A}},\forall a\in A,\;(a+v)+w=a+(v+w)}"> (here the last + is the addition in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79a63d1d5ad20924a9e908d0f805f2b16e745dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.647ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {A}}}">)</dd></dl></li> <li><a href="/wiki/Free_action" class="mw-redirect" title="Free action">Free</a> and <a href="/wiki/Transitive_action" class="mw-redirect" title="Transitive action">transitive action</a>: <dl><dd>For every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in A}">, the mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {A}}\to A\colon v\mapsto a+v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> <mo stretchy="false">→</mo> <mi>A</mi> <mo>:</mo> <mi>v</mi> <mo stretchy="false">↦</mo> <mi>a</mi> <mo>+</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {A}}\to A\colon v\mapsto a+v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/263fcca7383f4d2b57a93470c989f4f6a4aaf5c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.977ex; height:3.843ex;" alt="{\displaystyle {\overrightarrow {A}}\to A\colon v\mapsto a+v}"> is a <a href="/wiki/Bijection" title="Bijection">bijection</a>.</dd></dl></li></ol> The first two properties are simply defining properties of a (right) group action. The third property characterizes free and transitive actions, the <a href="/wiki/Onto" class="mw-redirect" title="Onto">onto</a> character coming from transitivity, and then the <a href="/wiki/Injective_function" title="Injective function">injective</a> character follows from the action being free. There is a fourth property that follows from 1, 2 above: <ol><li class="mw-empty-elt"></li><li value="4">Existence of one-to-one <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translations</a></li> <dl><dd>For all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in {\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\in {\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e55de186a95de885889df10b81e1d347aa1058f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.615ex; height:3.676ex;" alt="{\displaystyle v\in {\overrightarrow {A}}}">, the mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to A\colon a\mapsto a+v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>A</mi> <mo>:</mo> <mi>a</mi> <mo stretchy="false">↦</mo> <mi>a</mi> <mo>+</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to A\colon a\mapsto a+v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de5b5b16c1a28c1b35e443cdd58be9904302d54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.176ex; height:2.343ex;" alt="{\displaystyle A\to A\colon a\mapsto a+v}"> is a bijection.</dd></dl></ol> Property 3 is often used in the following equivalent form (the 5th property). <ol><li class="mw-empty-elt"></li><li value="5">Subtraction:</li> <dl><dd>For every a, b in A, there exists a unique <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in {\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\in {\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e55de186a95de885889df10b81e1d347aa1058f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.615ex; height:3.676ex;" alt="{\displaystyle v\in {\overrightarrow {A}}}">, denoted b – a, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=a+v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=a+v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0100ae48c18e183bef56cb0607d005c0f5b820b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.294ex; height:2.343ex;" alt="{\displaystyle b=a+v}">.</dd></dl></ol> Another way to express the definition is that an affine space is a <a href="/wiki/Principal_homogeneous_space" title="Principal homogeneous space">principal homogeneous space</a> for the action of the <a href="/wiki/Additive_group" title="Additive group">additive group</a> of a vector space. Homogeneous spaces are, by definition, endowed with a transitive group action, and for a principal homogeneous space, such a transitive action is, by definition, free. <div class="mw-heading mw-heading3"><h3 id="Subtraction_and_Weyl's_axioms">Subtraction and Weyl's axioms</h3>[<a href="/w/index.php?title=Affine_space&action=edit&section=3" title="Edit section: Subtraction and Weyl's axioms">edit</a>]</div> The properties of the group action allows for the definition of subtraction for any given ordered pair (b, a) of points in A, producing a vector of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79a63d1d5ad20924a9e908d0f805f2b16e745dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.647ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {A}}}">. This vector, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b-a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b-a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecca61f9c918fe1deb227ed79d4979d70c443ea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle b-a}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {ab}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>a</mi> <mi>b</mi> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {ab}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9f5228fc1782efa416fdce3110c0200ef7bcc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {ab}}}">, is defined to be the unique vector in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79a63d1d5ad20924a9e908d0f805f2b16e745dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.647ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {A}}}"> such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+(b-a)=b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+(b-a)=b.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d9b6fe8a03360c586b36ebff369555e40822d90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.69ex; height:2.843ex;" alt="{\displaystyle a+(b-a)=b.}"></dd></dl> Existence follows from the transitivity of the action, and uniqueness follows because the action is free. This subtraction has the two following properties, called <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a>'s axioms:<a href="#cite_note-7">[7]</a> <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall a\in A,\;\forall v\in {\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>∈</mo> <mi>A</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>v</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a\in A,\;\forall v\in {\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85a01a67376d7f7bfcde79ff7deb526a17ca11db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.693ex; height:4.009ex;" alt="{\displaystyle \forall a\in A,\;\forall v\in {\overrightarrow {A}}}">, there is a unique point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\in A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26329067b4b3630a2bb6f7e6b3ee367b70918371" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.581ex; height:2.176ex;" alt="{\displaystyle b\in A}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b-a=v.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo>=</mo> <mi>v</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b-a=v.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24264b1a0a5d4c764f65b84ed6549d5628b0e9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.941ex; height:2.343ex;" alt="{\displaystyle b-a=v.}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall a,b,c\in A,\;(c-b)+(b-a)=c-a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi>A</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <mo>−</mo> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a,b,c\in A,\;(c-b)+(b-a)=c-a.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ad4d9df1277d289d27d3a1a762b62deec9fcb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.051ex; height:2.843ex;" alt="{\displaystyle \forall a,b,c\in A,\;(c-b)+(b-a)=c-a.}"></li></ol> The <a href="/wiki/Equipollence_(geometry)#Parallelogram_property" title="Equipollence (geometry)">parallelogram property</a> is satisfied in affine spaces, where it is expressed as: given four points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c,d,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c,d,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec57da23813418a4c16caa6046dc2bf2f47806f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.199ex; height:2.509ex;" alt="{\displaystyle a,b,c,d,}"> the equalities <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b-a=d-c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo>=</mo> <mi>d</mi> <mo>−</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b-a=d-c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a8ee54905a3968f6835ea87dc1d2f4463774f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.229ex; height:2.343ex;" alt="{\displaystyle b-a=d-c}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c-a=d-b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>−</mo> <mi>a</mi> <mo>=</mo> <mi>d</mi> <mo>−</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c-a=d-b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8eb4e4074fe07023e2c8457b80a823600577657" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.229ex; height:2.343ex;" alt="{\displaystyle c-a=d-b}"> are equivalent. This results from the second Weyl's axiom, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d-a=(d-b)+(b-a)=(d-c)+(c-a).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>−</mo> <mi>a</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d-a=(d-b)+(b-a)=(d-c)+(c-a).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80277e7579caa6ad091db837c4bed15673819f89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.31ex; height:2.843ex;" alt="{\displaystyle d-a=(d-b)+(b-a)=(d-c)+(c-a).}"> Affine spaces can be equivalently defined as a point set A, together with a vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79a63d1d5ad20924a9e908d0f805f2b16e745dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.647ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {A}}}">, and a subtraction satisfying Weyl's axioms. In this case, the addition of a vector to a point is defined from the first of Weyl's axioms. <div class="mw-heading mw-heading2"><h2 id="Affine_subspaces_and_parallelism">Affine subspaces and parallelism</h2>[<a href="/w/index.php?title=Affine_space&action=edit&section=4" title="Edit section: Affine subspaces and parallelism">edit</a>]</div> An affine subspace (also called, in some contexts, a linear variety, a <a href="/wiki/Flat_(geometry)" title="Flat (geometry)">flat</a>, or, over the <a href="/wiki/Real_number" title="Real number">real numbers</a>, a linear manifold) B of an affine space A is a <a href="/wiki/Subset" title="Subset">subset</a> of A such that, given a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/739c140e9ae7f2a90119c5b40507d1ad2bc3396b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.834ex; height:2.176ex;" alt="{\displaystyle a\in B}">, the set of vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {B}}=\{b-a\mid b\in B\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo>∣</mo> <mi>b</mi> <mo>∈</mo> <mi>B</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {B}}=\{b-a\mid b\in B\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fbc5b3170b8fe3e0ef2081944e07481aada02cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.548ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {B}}=\{b-a\mid b\in B\}}"> is a <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79a63d1d5ad20924a9e908d0f805f2b16e745dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.647ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {A}}}">. This property, which does not depend on the choice of a, implies that B is an affine space, which has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31fb3faaa4a9e6eb6abcef4a41c4bbeb84cc3c90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.517ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {B}}}"> as its associated vector space. The affine subspaces of A are the subsets of A of the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+V=\{a+w:w\in V\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>V</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>+</mo> <mi>w</mi> <mo>:</mo> <mi>w</mi> <mo>∈</mo> <mi>V</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+V=\{a+w:w\in V\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e19c7ed4cbcabb6fe6b34c19717dae4ad0efadc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.891ex; height:2.843ex;" alt="{\displaystyle a+V=\{a+w:w\in V\},}"></dd></dl> where a is a point of A, and V a linear subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79a63d1d5ad20924a9e908d0f805f2b16e745dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.647ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {A}}}">. The linear subspace associated with an affine subspace is often called its <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style>direction, and two subspaces that share the same direction are said to be parallel. This implies the following generalization of <a href="/wiki/Playfair%27s_axiom" title="Playfair's axiom">Playfair's axiom</a>: Given a direction V, for any point a of A there is one and only one affine subspace of direction V, which passes through a, namely the subspace a + V. Every translation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to A:a\mapsto a+v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>A</mi> <mo>:</mo> <mi>a</mi> <mo stretchy="false">↦</mo> <mi>a</mi> <mo>+</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to A:a\mapsto a+v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a75490bd2c00651d0ad38f687eb5c3378382b8ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.079ex; height:2.343ex;" alt="{\displaystyle A\to A:a\mapsto a+v}"> maps any affine subspace to a parallel subspace. The term parallel is also used for two affine subspaces such that the direction of one is included in the direction of the other. <div class="mw-heading mw-heading2"><h2 id="Affine_map">Affine map</h2>[<a href="/w/index.php?title=Affine_space&action=edit&section=5" title="Edit section: Affine map">edit</a>]</div> Given two affine spaces A and B whose associated vector spaces are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79a63d1d5ad20924a9e908d0f805f2b16e745dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.647ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {A}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31fb3faaa4a9e6eb6abcef4a41c4bbeb84cc3c90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.517ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {B}}}">, an <a href="/wiki/Affine_map" class="mw-redirect" title="Affine map">affine map</a> or affine homomorphism from A to B is a map <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20040a52d9391f2fe271f0aaa300bf7887a0c7b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.337ex; height:2.509ex;" alt="{\displaystyle f:A\to B}"></dd></dl> such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\overrightarrow {f}}:{\overrightarrow {A}}&\to {\overrightarrow {B}}\\b-a&\mapsto f(b)-f(a)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>→</mo> </mover> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo>→</mo> </mover> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overrightarrow {f}}:{\overrightarrow {A}}&\to {\overrightarrow {B}}\\b-a&\mapsto f(b)-f(a)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7bc34f1be10b1d537bfb94c48f7aa7c6187eb69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.995ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}{\overrightarrow {f}}:{\overrightarrow {A}}&\to {\overrightarrow {B}}\\b-a&\mapsto f(b)-f(a)\end{aligned}}}"></dd></dl> is a <a href="/wiki/Well_defined" class="mw-redirect" title="Well defined">well defined</a> linear map. By <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> being well defined is meant that b – a = d – c implies f(b) – f(a) = f(d) – f(c). This implies that, for a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in A}"> and a vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in {\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\in {\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e55de186a95de885889df10b81e1d347aa1058f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.615ex; height:3.676ex;" alt="{\displaystyle v\in {\overrightarrow {A}}}">, one has <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a+v)=f(a)+{\overrightarrow {f}}(v).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>→</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(a+v)=f(a)+{\overrightarrow {f}}(v).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/629be545e80a2dd87495904921cbb96fe9436de6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.928ex; height:4.176ex;" alt="{\displaystyle f(a+v)=f(a)+{\overrightarrow {f}}(v).}"></dd></dl> Therefore, since for any given b in A, b = a + v for a unique v, f is completely defined by its value on a single point and the associated linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfa88ab873934603aafa809d533b4c49fc16adda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.802ex; height:4.009ex;" alt="{\displaystyle {\overrightarrow {f}}}">. <div class="mw-heading mw-heading3"><h3 id="Endomorphisms">Endomorphisms</h3>[<a href="/w/index.php?title=Affine_space&action=edit&section=6" title="Edit section: Endomorphisms">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Affine_transformation" title="Affine transformation">Affine transformation</a> and <a href="/wiki/Affine_group" title="Affine group">Affine group</a></div> An affine transformation or <a href="/wiki/Endomorphism" title="Endomorphism">endomorphism</a> of an affine space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is an affine map from that space to itself. One important <a href="/wiki/Family_of_sets" title="Family of sets">family</a> of examples is the translations: given a vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdbb1a68c861cbd0cfda4f71510f67eed27c7cb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:3.009ex;" alt="{\displaystyle {\overrightarrow {v}}}">, the translation map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\overrightarrow {v}}:A\rightarrow A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo>→</mo> </mover> </mrow> </msub> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\overrightarrow {v}}:A\rightarrow A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ebeca4fbe60527653f977b904f92b489d43e3aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:12.335ex; height:3.343ex;" alt="{\displaystyle T_{\overrightarrow {v}}:A\rightarrow A}"> that sends <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\mapsto a+{\overrightarrow {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">↦</mo> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\mapsto a+{\overrightarrow {v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c26f8b6adca71bbe522d3f9eb81e16ed0ee33b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.302ex; height:3.176ex;" alt="{\displaystyle a\mapsto a+{\overrightarrow {v}}}"> for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is an affine map. Another important family of examples are the linear maps centred at an origin: given a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> and a linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">, one may define an affine map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{M,b}:A\rightarrow A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{M,b}:A\rightarrow A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad196ea8e4e01e4016de27e2a97775f1dc820ab5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.742ex; height:2.843ex;" alt="{\displaystyle L_{M,b}:A\rightarrow A}"> by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{M,b}(a)=b+M(a-b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{M,b}(a)=b+M(a-b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a729b56b1e43dfe0281b42b5fc4644f671a87948" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24ex; height:3.009ex;" alt="{\displaystyle L_{M,b}(a)=b+M(a-b)}"> for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. After making a choice of origin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">, any affine map may be written uniquely as a combination of a translation and a linear map centred at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">. <div class="mw-heading mw-heading2"><h2 id="Vector_spaces_as_affine_spaces">Vector spaces as affine spaces</h2>[<a href="/w/index.php?title=Affine_space&action=edit&section=7" title="Edit section: Vector spaces as affine spaces">edit</a>]</div> Every vector space V may be considered as an affine space over itself. This means that every element of V may be considered either as a point or as a vector. This affine space is sometimes denoted (V, V) for emphasizing the double role of the elements of V. When considered as a point, the <a href="/wiki/Zero_vector" class="mw-redirect" title="Zero vector">zero vector</a> is commonly denoted o (or O, when upper-case letters are used for points) and called the origin. If A is another affine space over the same vector space (that is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a59e4ac6e1149d31ede7eaf0a7ec295f0f6480bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.532ex; height:3.676ex;" alt="{\displaystyle V={\overrightarrow {A}}}">) the choice of any point a in A defines a unique affine isomorphism, which is the identity of V and maps a to o. In other words, the choice of an origin a in A allows us to identify A and (V, V) <a href="/wiki/Up_to" title="Up to">up to</a> a <a href="/wiki/Canonical_isomorphism" class="mw-redirect" title="Canonical isomorphism">canonical isomorphism</a>. The counterpart of this property is that the affine space A may be identified with the vector space V in which "the place of the origin has been forgotten". <div class="mw-heading mw-heading2"><h2 id="Relation_to_Euclidean_spaces">Relation to Euclidean spaces</h2>[<a href="/w/index.php?title=Affine_space&action=edit&section=8" title="Edit section: Relation to Euclidean spaces">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Definition_of_Euclidean_spaces">Definition of Euclidean spaces</h3>[<a href="/w/index.php?title=Affine_space&action=edit&section=9" title="Edit section: Definition of Euclidean spaces">edit</a>]</div> <a href="/wiki/Euclidean_spaces" class="mw-redirect" title="Euclidean spaces">Euclidean spaces</a> (including the one-dimensional line, two-dimensional plane, and three-dimensional space commonly studied in elementary geometry, as well as higher-dimensional analogues) are affine spaces. Indeed, in most modern definitions, a Euclidean space is defined to be an affine space, such that the associated vector space is a real <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a> of finite dimension, that is a vector space over the reals with a <a href="/wiki/Positive-definite_quadratic_form" class="mw-redirect" title="Positive-definite quadratic form">positive-definite quadratic form</a> q(x). The inner product of two vectors x and y is the value of the <a href="/wiki/Symmetric_bilinear_form" title="Symmetric bilinear form">symmetric bilinear form</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot y={\frac {1}{2}}(q(x+y)-q(x)-q(y)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot y={\frac {1}{2}}(q(x+y)-q(x)-q(y)).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa3faeb3da613de6bd56dde869bae11ec7c831f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.846ex; height:5.176ex;" alt="{\displaystyle x\cdot y={\frac {1}{2}}(q(x+y)-q(x)-q(y)).}"></dd></dl> The usual <a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a> between two points A and B is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,B)={\sqrt {q(B-A)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>q</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo>−</mo> <mi>A</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(A,B)={\sqrt {q(B-A)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0bc59fc12a8f94498ce0eb52326f37fd6bff2b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.862ex; height:4.843ex;" alt="{\displaystyle d(A,B)={\sqrt {q(B-A)}}.}"></dd></dl> In older definition of Euclidean spaces through <a href="/wiki/Synthetic_geometry" title="Synthetic geometry">synthetic geometry</a>, vectors are defined as <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> of <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pairs</a> of points under <a href="/wiki/Equipollence_(geometry)" title="Equipollence (geometry)">equipollence</a> (the pairs (A, B) and (C, D) are equipollent if the points A, B, D, C (in this order) form a <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a>). It is straightforward to verify that the vectors form a vector space, the square of the <a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a> is a quadratic form on the space of vectors, and the two definitions of Euclidean spaces are equivalent. <div class="mw-heading mw-heading3"><h3 id="Affine_properties">Affine properties</h3>[<a href="/w/index.php?title=Affine_space&action=edit&section=10" title="Edit section: Affine properties">edit</a>]</div> In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, the common phrase "affine property" refers to a property that can be proved in affine spaces, that is, it can be proved without using the quadratic form and its associated inner product. In other words, an affine property is a property that does not involve lengths and angles. Typical examples are <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallelism</a>, and the definition of a <a href="/wiki/Tangent" title="Tangent">tangent</a>. A non-example is the definition of a <a href="/wiki/Normal_(geometry)" title="Normal (geometry)">normal</a>. Equivalently, an affine property is a property that is invariant under <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformations</a> of the Euclidean space. <div class="mw-heading mw-heading2"><h2 id="Affine_combinations_and_barycenter">Affine combinations and barycenter</h2>[<a href="/w/index.php?title=Affine_space&action=edit&section=11" title="Edit section: Affine combinations and barycenter">edit</a>]</div> Let a1, ..., an be a collection of n points in an affine space, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1},\dots ,\lambda _{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> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1},\dots ,\lambda _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0980b08cb9c0fae51d9109a1c6319b22807e0eea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.161ex; height:2.509ex;" alt="{\displaystyle \lambda _{1},\dots ,\lambda _{n}}"> be n elements of the <a href="/wiki/Ground_field" title="Ground field">ground field</a>. Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1}+\dots +\lambda _{n}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1}+\dots +\lambda _{n}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbd693cec3ff50642341afcdec0b6164d025772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.648ex; height:2.509ex;" alt="{\displaystyle \lambda _{1}+\dots +\lambda _{n}=0}">. For any two points o and o' one has <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1}{\overrightarrow {oa_{1}}}+\dots +\lambda _{n}{\overrightarrow {oa_{n}}}=\lambda _{1}{\overrightarrow {o'a_{1}}}+\dots +\lambda _{n}{\overrightarrow {o'a_{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> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>o</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>o</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>o</mi> <mo>′</mo> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>o</mi> <mo>′</mo> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1}{\overrightarrow {oa_{1}}}+\dots +\lambda _{n}{\overrightarrow {oa_{n}}}=\lambda _{1}{\overrightarrow {o'a_{1}}}+\dots +\lambda _{n}{\overrightarrow {o'a_{n}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3bc06cbbf080f47ee95cdae92f80f353f374efc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.491ex; width:46.385ex; height:4.509ex;" alt="{\displaystyle \lambda _{1}{\overrightarrow {oa_{1}}}+\dots +\lambda _{n}{\overrightarrow {oa_{n}}}=\lambda _{1}{\overrightarrow {o'a_{1}}}+\dots +\lambda _{n}{\overrightarrow {o'a_{n}}}.}"></dd></dl> Thus, this sum is independent of the choice of the origin, and the resulting vector may be denoted <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1}a_{1}+\dots +\lambda _{n}a_{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> <mi>a</mi> <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> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1}a_{1}+\dots +\lambda _{n}a_{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b24e4266b1eef98014cc1082858c4dfda0cb3e48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.766ex; height:2.509ex;" alt="{\displaystyle \lambda _{1}a_{1}+\dots +\lambda _{n}a_{n}.}"></dd></dl> When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2,\lambda _{1}=1,\lambda _{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2,\lambda _{1}=1,\lambda _{2}=-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6338317c67bb6db345ce2f5932f0440110f543e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.872ex; height:2.509ex;" alt="{\displaystyle n=2,\lambda _{1}=1,\lambda _{2}=-1}">, one retrieves the definition of the subtraction of points. Now suppose instead that the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> elements satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1}+\dots +\lambda _{n}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1}+\dots +\lambda _{n}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9645ebdd7fbf5f93058a0c9df6f238c9a86ee8a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.648ex; height:2.509ex;" alt="{\displaystyle \lambda _{1}+\dots +\lambda _{n}=1}">. For some choice of an origin o, denote by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> the unique point such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1}{\overrightarrow {oa_{1}}}+\dots +\lambda _{n}{\overrightarrow {oa_{n}}}={\overrightarrow {og}}.}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>o</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>o</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>o</mi> <mi>g</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1}{\overrightarrow {oa_{1}}}+\dots +\lambda _{n}{\overrightarrow {oa_{n}}}={\overrightarrow {og}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d47689c3c5a1232cd16482a3fbfef325148197e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.342ex; width:26.709ex; height:3.509ex;" alt="{\displaystyle \lambda _{1}{\overrightarrow {oa_{1}}}+\dots +\lambda _{n}{\overrightarrow {oa_{n}}}={\overrightarrow {og}}.}"></dd></dl> One can show that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> is independent from the choice of o. Therefore, if <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1}+\dots +\lambda _{n}=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1}+\dots +\lambda _{n}=1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb7694543f0ac5b30a262746c049323873b149b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.295ex; height:2.509ex;" alt="{\displaystyle \lambda _{1}+\dots +\lambda _{n}=1,}"></dd></dl> one may write <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=\lambda _{1}a_{1}+\dots +\lambda _{n}a_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <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> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=\lambda _{1}a_{1}+\dots +\lambda _{n}a_{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab52f0ade1d1db4cd27e40baa6caab3c9b6a96e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.981ex; height:2.509ex;" alt="{\displaystyle g=\lambda _{1}a_{1}+\dots +\lambda _{n}a_{n}.}"></dd></dl> The point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> is called the <a href="/wiki/Centroid" title="Centroid">barycenter</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"> for the weights <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72fde940918edf84caf3d406cc7d31949166820f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}}">. One says also that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> is an <a href="/wiki/Affine_combination" title="Affine combination">affine combination</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"> with <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72fde940918edf84caf3d406cc7d31949166820f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}}">. <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Affine_space&action=edit&section=12" title="Edit section: Examples">edit</a>]</div> <ul><li>When children find the answers to sums such as 4 + 3 or 4 − 2 by counting right or left on a <a href="/wiki/Number_line" title="Number line">number line</a>, they are treating the number line as a one-dimensional affine space.</li> <li><a href="/wiki/Time" title="Time">Time</a> can be modelled as a one-dimensional affine space. Specific points in time (such as a date on the calendar) are points in the affine space, while durations (such as a number of days) are displacements.</li> <li>The space of energies is an affine space for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">, since it is often not meaningful to talk about absolute energy, but it is meaningful to talk about energy differences. The <a href="/wiki/Vacuum_energy" title="Vacuum energy">vacuum energy</a> when it is defined picks out a canonical origin.</li> <li><a href="/wiki/Physical_space" class="mw-redirect" title="Physical space">Physical space</a> is often modelled as an affine space for <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><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}}"> in non-relativistic settings and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{1,3}}"> <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>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{1,3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97bbc96ea97da9adfb87ac79a78c8e604c97cd7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.012ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{1,3}}"> in the relativistic setting. To distinguish them from the vector space these are sometimes called <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{E}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>E</mtext> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{E}}(3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b12266f999d95c9a711e3f5a00692bfb66f6ab6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle {\text{E}}(3)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{E}}(1,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>E</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{E}}(1,3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6762e6aed67f62945a586c7e5a8b2894d2f56765" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.751ex; height:2.843ex;" alt="{\displaystyle {\text{E}}(1,3)}">.</li> <li>Any <a href="/wiki/Coset" title="Coset">coset</a> of a subspace V of a vector space is an affine space over that subspace.</li> <li>In particular, a line in the plane that doesn't pass through the origin is an affine space that is not a vector space relative to the operations it inherits from <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><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}}">, although it can be given a canonical vector space structure by picking the point closest to the origin as the zero vector; likewise in higher dimensions and for any normed vector space</li> <li>If T is a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> and b lies in its <a href="/wiki/Column_space" class="mw-redirect" title="Column space">column space</a>, the set of solutions of the equation Tx = b is an affine space over the subspace of solutions of Tx = 0.</li> <li>The solutions of an inhomogeneous <a href="/wiki/Linear_differential_equation" title="Linear differential equation">linear differential equation</a> form an affine space over the solutions of the corresponding homogeneous linear equation.</li> <li>Generalizing all of the above, if T : V → W is a linear map and y lies in its <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a>, the set of solutions x ∈ V to the equation Tx = y is a coset of the kernel of T , and is therefore an affine space over Ker T .</li> <li>The space of (linear) <a href="/wiki/Complementary_subspaces" class="mw-redirect" title="Complementary subspaces">complementary subspaces</a> of a vector subspace V in a vector space W is an affine space, over Hom(W/V, V). That is, if 0 → V → W → X → 0 is a <a href="/wiki/Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequence</a> of vector spaces, then the space of all <a href="/wiki/Split_exact_sequence" title="Split exact sequence">splittings</a> of the exact sequence naturally carries the structure of an affine space over Hom(X, V).</li> <li>The space of <a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">connections</a> (viewed from the <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\xrightarrow {\pi } M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mi>π</mi> </mpadded> </mover> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\xrightarrow {\pi } M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f4fbd0716bcabdbbce17a99087bc4478bb4096a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-top: -0.409ex; width:7.832ex; height:3.343ex;" alt="{\displaystyle E\xrightarrow {\pi } M}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is a <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a>) is an affine space for the vector space of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{End}}(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>End</mtext> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{End}}(E)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3440279a0745bee2ce96088e2e39bd04d59bb581" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.753ex; height:2.843ex;" alt="{\displaystyle {\text{End}}(E)}"> valued <a href="/wiki/1-forms" class="mw-redirect" title="1-forms">1-forms</a>. The space of connections (viewed from the <a href="/wiki/Principal_bundle" title="Principal bundle">principal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\xrightarrow {\pi } M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mi>π</mi> </mpadded> </mover> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\xrightarrow {\pi } M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e293f6c88af4af28d5737387bfaa158d3873650" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-top: -0.409ex; width:7.802ex; height:3.343ex;" alt="{\displaystyle P\xrightarrow {\pi } M}">) is an affine space for the vector space of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ad}}(P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>ad</mtext> </mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{ad}}(P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19e76680c5beccb807dc345bad4747ad0fb48349" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.01ex; height:2.843ex;" alt="{\displaystyle {\text{ad}}(P)}">-valued 1-forms, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ad}}(P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>ad</mtext> </mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{ad}}(P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19e76680c5beccb807dc345bad4747ad0fb48349" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.01ex; height:2.843ex;" alt="{\displaystyle {\text{ad}}(P)}"> is the <a href="/wiki/Associated_vector_bundle" class="mw-redirect" title="Associated vector bundle">associated</a> <a href="/wiki/Adjoint_bundle" title="Adjoint bundle">adjoint bundle</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Affine_span_and_bases">Affine span and bases</h2>[<a href="/w/index.php?title=Affine_space&action=edit&section=13" title="Edit section: Affine span and bases">edit</a>]</div> For any non-empty subset X of an affine space A, there is a smallest affine subspace that contains it, called the affine span of X. It is the intersection of all affine subspaces containing X, and its direction is the intersection of the directions of the affine subspaces that contain X. The affine span of X is the set of all (finite) affine combinations of points of X, and its direction is the <a href="/wiki/Linear_span" title="Linear span">linear span</a> of the x − y for x and y in X. If one chooses a particular point x0, the direction of the affine span of X is also the linear span of the x – x0 for x in X. One says also that the affine span of X is generated by X and that X is a generating set of its affine span. A set X of points of an affine space is said to be <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">affinely independent or, simply, independent, if the affine span of any <a href="/wiki/Strict_subset" class="mw-redirect" title="Strict subset">strict subset</a> of X is a strict subset of the affine span of X. An <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">affine basis or barycentric frame (see <a href="#Barycentric_coordinates">§ Barycentric coordinates</a>, below) of an affine space is a generating set that is also independent (that is a <a href="/wiki/Minimal_generating_set" class="mw-redirect" title="Minimal generating set">minimal generating set</a>). Recall that the dimension of an affine space is the dimension of its associated vector space. The bases of an affine space of finite dimension n are the independent subsets of n + 1 elements, or, equivalently, the generating subsets of n + 1 elements. Equivalently, {x0, ..., xn} is an affine basis of an affine space <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> {x1 − x0, ..., xn − x0} is a <a href="/wiki/Linear_basis" class="mw-redirect" title="Linear basis">linear basis</a> of the associated vector space. <div class="mw-heading mw-heading2"><h2 id="Coordinates">Coordinates</h2>[<a href="/w/index.php?title=Affine_space&action=edit&section=14" title="Edit section: Coordinates">edit</a>]</div> There are two strongly related kinds of <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate systems</a> that may be defined on affine spaces. <div class="mw-heading mw-heading3"><h3 id="Barycentric_coordinates">Barycentric coordinates</h3>[<a href="/w/index.php?title=Affine_space&action=edit&section=15" title="Edit section: Barycentric coordinates">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Barycentric_coordinate_system" title="Barycentric coordinate system">Barycentric coordinate system</a></div> Let A be an affine space of dimension n over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> k, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x_{0},\dots ,x_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x_{0},\dots ,x_{n}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/007e042ff4cda39d85d58e1876cda96a1e0658c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.435ex; height:2.843ex;" alt="{\displaystyle \{x_{0},\dots ,x_{n}\}}"> be an affine basis of A. The properties of an affine basis imply that for every x in A there is a unique (n + 1)-<a href="/wiki/Tuple" title="Tuple">tuple</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda _{0},\dots ,\lambda _{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>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda _{0},\dots ,\lambda _{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fa2b2dfbd78a8dda94fe7d019b04dc72cac1985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.971ex; height:2.843ex;" alt="{\displaystyle (\lambda _{0},\dots ,\lambda _{n})}"> of elements of k such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{0}+\dots +\lambda _{n}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{0}+\dots +\lambda _{n}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4410120f23a1e884b11a16847ccdbfd6ecb505" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.648ex; height:2.509ex;" alt="{\displaystyle \lambda _{0}+\dots +\lambda _{n}=1}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\lambda _{0}x_{0}+\dots +\lambda _{n}x_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\lambda _{0}x_{0}+\dots +\lambda _{n}x_{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c30ca97ef4b3771539ab3ed62bbe40c45e36851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.394ex; height:2.509ex;" alt="{\displaystyle x=\lambda _{0}x_{0}+\dots +\lambda _{n}x_{n}.}"></dd></dl> The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72fde940918edf84caf3d406cc7d31949166820f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}}"> are called the barycentric coordinates of x over the affine basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x_{0},\dots ,x_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x_{0},\dots ,x_{n}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/007e042ff4cda39d85d58e1876cda96a1e0658c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.435ex; height:2.843ex;" alt="{\displaystyle \{x_{0},\dots ,x_{n}\}}">. If the xi are viewed as bodies that have weights (or masses) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72fde940918edf84caf3d406cc7d31949166820f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}}">, the point x is thus the <a href="/wiki/Centroid" title="Centroid">barycenter</a> of the xi, and this explains the origin of the term barycentric coordinates. The barycentric coordinates define an affine isomorphism between the affine space A and the affine subspace of kn + 1 defined by the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{0}+\dots +\lambda _{n}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{0}+\dots +\lambda _{n}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4410120f23a1e884b11a16847ccdbfd6ecb505" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.648ex; height:2.509ex;" alt="{\displaystyle \lambda _{0}+\dots +\lambda _{n}=1}">. For affine spaces of infinite dimension, the same definition applies, using only finite sums. This means that for each point, only a finite number of coordinates are non-zero. <div class="mw-heading mw-heading3"><h3 id="Affine_coordinates">Affine coordinates</h3>[<a href="/w/index.php?title=Affine_space&action=edit&section=16" title="Edit section: Affine coordinates">edit</a>]</div> An affine frame is a <a href="/wiki/Coordinate_frame" class="mw-redirect" title="Coordinate frame">coordinate frame</a> of an affine space, consisting of a point, called the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a>, and a <a href="/wiki/Linear_basis" class="mw-redirect" title="Linear basis">linear basis</a> of the associated vector space. More precisely, for an affine space A with associated vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79a63d1d5ad20924a9e908d0f805f2b16e745dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.647ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {A}}}">, the origin o belongs to A, and the linear basis is a basis (v1, ..., vn) of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79a63d1d5ad20924a9e908d0f805f2b16e745dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.647ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {A}}}"> (for simplicity of the notation, we consider only the case of finite dimension, the general case is similar). For each point p of A, there is a unique sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1},\dots ,\lambda _{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> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1},\dots ,\lambda _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0980b08cb9c0fae51d9109a1c6319b22807e0eea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.161ex; height:2.509ex;" alt="{\displaystyle \lambda _{1},\dots ,\lambda _{n}}"> of elements of the ground field such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=o+\lambda _{1}v_{1}+\dots +\lambda _{n}v_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>o</mi> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <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> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=o+\lambda _{1}v_{1}+\dots +\lambda _{n}v_{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a241f4da37377a89d62ab447d54cd108ae6abc5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:26.887ex; height:2.509ex;" alt="{\displaystyle p=o+\lambda _{1}v_{1}+\dots +\lambda _{n}v_{n},}"></dd></dl> or equivalently <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {op}}=\lambda _{1}v_{1}+\dots +\lambda _{n}v_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>o</mi> <mi>p</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <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> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {op}}=\lambda _{1}v_{1}+\dots +\lambda _{n}v_{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73ab04c0b95b682d852ba8321591ee77e56c31a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.984ex; height:3.343ex;" alt="{\displaystyle {\overrightarrow {op}}=\lambda _{1}v_{1}+\dots +\lambda _{n}v_{n}.}"></dd></dl> The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72fde940918edf84caf3d406cc7d31949166820f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}}"> are called the affine coordinates of p over the affine frame (o, v1, ..., vn). Example: In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> are affine coordinates relative to an <a href="/wiki/Orthonormal_frame_(Euclidean_geometry)" class="mw-redirect" title="Orthonormal frame (Euclidean geometry)">orthonormal frame</a>, that is an affine frame (o, v1, ..., vn) such that (v1, ..., vn) is an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a>. <div class="mw-heading mw-heading3"><h3 id="Relationship_between_barycentric_and_affine_coordinates">Relationship between barycentric and affine coordinates</h3>[<a href="/w/index.php?title=Affine_space&action=edit&section=17" title="Edit section: Relationship between barycentric and affine coordinates">edit</a>]</div> Barycentric coordinates and affine coordinates are strongly related, and may be considered as equivalent. In fact, given a barycentric frame <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},\dots ,x_{n}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},\dots ,x_{n}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe79334b26a058149d8c84b726460377475e8bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.566ex; height:2.843ex;" alt="{\displaystyle (x_{0},\dots ,x_{n}),}"></dd></dl> one deduces immediately the affine frame <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},{\overrightarrow {x_{0}x_{1}}},\dots ,{\overrightarrow {x_{0}x_{n}}})=\left(x_{0},x_{1}-x_{0},\dots ,x_{n}-x_{0}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},{\overrightarrow {x_{0}x_{1}}},\dots ,{\overrightarrow {x_{0}x_{n}}})=\left(x_{0},x_{1}-x_{0},\dots ,x_{n}-x_{0}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feb68180edc027a8c75c523530b68873e3140b6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-top: -0.342ex; width:50.284ex; height:3.676ex;" alt="{\displaystyle (x_{0},{\overrightarrow {x_{0}x_{1}}},\dots ,{\overrightarrow {x_{0}x_{n}}})=\left(x_{0},x_{1}-x_{0},\dots ,x_{n}-x_{0}\right),}"></dd></dl> and, if <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\lambda _{0},\lambda _{1},\dots ,\lambda _{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>λ</mi> <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> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\lambda _{0},\lambda _{1},\dots ,\lambda _{n}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e976f8d679c43f97c3b6652c83e9686d4735d70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.414ex; height:2.843ex;" alt="{\displaystyle \left(\lambda _{0},\lambda _{1},\dots ,\lambda _{n}\right)}"></dd></dl> are the barycentric coordinates of a point over the barycentric frame, then the affine coordinates of the same point over the affine frame are <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\lambda _{1},\dots ,\lambda _{n}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>λ</mi> <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> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\lambda _{1},\dots ,\lambda _{n}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c5d8b5824a7f53bea98c66039b126a5f1ea00a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.005ex; height:2.843ex;" alt="{\displaystyle \left(\lambda _{1},\dots ,\lambda _{n}\right).}"></dd></dl> Conversely, if <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(o,v_{1},\dots ,v_{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>o</mi> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(o,v_{1},\dots ,v_{n}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b48eb98647af912da0588284ecbf32ba23de4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.677ex; height:2.843ex;" alt="{\displaystyle \left(o,v_{1},\dots ,v_{n}\right)}"></dd></dl> is an affine frame, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(o,o+v_{1},\dots ,o+v_{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>o</mi> <mo>,</mo> <mi>o</mi> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>o</mi> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(o,o+v_{1},\dots ,o+v_{n}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed716e47f7868a9cc9ef023561bfc06c0b8a4e8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.613ex; height:2.843ex;" alt="{\displaystyle \left(o,o+v_{1},\dots ,o+v_{n}\right)}"></dd></dl> is a barycentric frame. If <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\lambda _{1},\dots ,\lambda _{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>λ</mi> <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> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\lambda _{1},\dots ,\lambda _{n}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56f7742a9f00ba87be1ef8a42c1af60f4b6b8fb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.971ex; height:2.843ex;" alt="{\displaystyle \left(\lambda _{1},\dots ,\lambda _{n}\right)}"></dd></dl> are the affine coordinates of a point over the affine frame, then its barycentric coordinates over the barycentric frame are <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(1-\lambda _{1}-\dots -\lambda _{n},\lambda _{1},\dots ,\lambda _{n}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>λ</mi> <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> <mo>,</mo> <msub> <mi>λ</mi> <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> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1-\lambda _{1}-\dots -\lambda _{n},\lambda _{1},\dots ,\lambda _{n}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d98f638ab4ee7de5cbd47586cacc4b4362d23507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.429ex; height:2.843ex;" alt="{\displaystyle \left(1-\lambda _{1}-\dots -\lambda _{n},\lambda _{1},\dots ,\lambda _{n}\right).}"></dd></dl> Therefore, barycentric and affine coordinates are almost equivalent. In most applications, affine coordinates are preferred, as involving less coordinates that are independent. However, in the situations where the important points of the studied problem are affinely independent, barycentric coordinates may lead to simpler computation, as in the following example. <div class="mw-heading mw-heading4"><h4 id="Example_of_the_triangle">Example of the triangle</h4>[<a href="/w/index.php?title=Affine_space&action=edit&section=18" title="Edit section: Example of the triangle">edit</a>]</div> The vertices of a non-flat <a href="/wiki/Triangle" title="Triangle">triangle</a> form an affine basis of the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a>. The barycentric coordinates allows easy characterization of the elements of the triangle that do not involve angles or distances: The vertices are the points of barycentric coordinates (1, 0, 0), (0, 1, 0) and (0, 0, 1). The lines supporting the <a href="/wiki/Edge_(geometry)" title="Edge (geometry)">edges</a> are the points that have a zero coordinate. The edges themselves are the points that have one zero coordinate and two nonnegative coordinates. The <a href="/wiki/Interior_(mathematics)" class="mw-redirect" title="Interior (mathematics)">interior</a> of the triangle are the points whose coordinates are all positive. The <a href="/wiki/Median_(geometry)" title="Median (geometry)">medians</a> are the points that have two equal coordinates, and the <a href="/wiki/Centroid" title="Centroid">centroid</a> is the point of coordinates (<style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.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>⁠1/3⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3⁠). <div class="mw-heading mw-heading3"><h3 id="Change_of_coordinates">Change of coordinates</h3>[<a href="/w/index.php?title=Affine_space&action=edit&section=19" title="Edit section: Change of coordinates">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Case_of_barycentric_coordinates">Case of barycentric coordinates</h4>[<a href="/w/index.php?title=Affine_space&action=edit&section=20" title="Edit section: Case of barycentric coordinates">edit</a>]</div> Barycentric coordinates are readily changed from one basis to another. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x_{0},\dots ,x_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x_{0},\dots ,x_{n}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/007e042ff4cda39d85d58e1876cda96a1e0658c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.435ex; height:2.843ex;" alt="{\displaystyle \{x_{0},\dots ,x_{n}\}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x'_{0},\dots ,x'_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x'_{0},\dots ,x'_{n}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2de731d0e287456709c7aa37233389adcb07695a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.435ex; height:3.009ex;" alt="{\displaystyle \{x'_{0},\dots ,x'_{n}\}}"> be affine bases of A. For every x in A there is some tuple <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\lambda _{0},\dots ,\lambda _{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\lambda _{0},\dots ,\lambda _{n}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37257cd716b800f1eb19c821cbbea5110fc322a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.486ex; height:2.843ex;" alt="{\displaystyle \{\lambda _{0},\dots ,\lambda _{n}\}}"> for which <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\lambda _{0}x_{0}+\dots +\lambda _{n}x_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\lambda _{0}x_{0}+\dots +\lambda _{n}x_{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c30ca97ef4b3771539ab3ed62bbe40c45e36851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.394ex; height:2.509ex;" alt="{\displaystyle x=\lambda _{0}x_{0}+\dots +\lambda _{n}x_{n}.}"></dd></dl> Similarly, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\in \{x_{0},\dots ,x_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}\in \{x_{0},\dots ,x_{n}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea22665da1292f62caa3b6a80050f01bb161069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.405ex; height:2.843ex;" alt="{\displaystyle x_{i}\in \{x_{0},\dots ,x_{n}\}}"> from the first basis, we now have in the second basis <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=\lambda _{i,0}x'_{0}+\dots +\lambda _{i,j}x'_{j}+\dots +\lambda _{i,n}x'_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}=\lambda _{i,0}x'_{0}+\dots +\lambda _{i,j}x'_{j}+\dots +\lambda _{i,n}x'_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/442ca185265086def20bd500494932469b296088" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:39.53ex; height:3.176ex;" alt="{\displaystyle x_{i}=\lambda _{i,0}x'_{0}+\dots +\lambda _{i,j}x'_{j}+\dots +\lambda _{i,n}x'_{n}}"></dd></dl> for some tuple <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\lambda _{i,0},\dots ,\lambda _{i,n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\lambda _{i,0},\dots ,\lambda _{i,n}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dde36be9985cdfc1e2b604446d3cd42f8f6c17a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.536ex; height:3.009ex;" alt="{\displaystyle \{\lambda _{i,0},\dots ,\lambda _{i,n}\}}">. Now we can rewrite our expression in the first basis as one in the second with <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,x=\sum _{i=0}^{n}\lambda _{i}x_{i}=\sum _{i=0}^{n}\lambda _{i}\sum _{j=0}^{n}\lambda _{i,j}x'_{j}=\sum _{j=0}^{n}{\biggl (}\sum _{i=0}^{n}\lambda _{i}\lambda _{i,j}{\biggr )}x'_{j}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>x</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,x=\sum _{i=0}^{n}\lambda _{i}x_{i}=\sum _{i=0}^{n}\lambda _{i}\sum _{j=0}^{n}\lambda _{i,j}x'_{j}=\sum _{j=0}^{n}{\biggl (}\sum _{i=0}^{n}\lambda _{i}\lambda _{i,j}{\biggr )}x'_{j}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d139fc3ec1de4f87b482712c2dac75c4506e4ec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:54.604ex; height:7.176ex;" alt="{\displaystyle \,x=\sum _{i=0}^{n}\lambda _{i}x_{i}=\sum _{i=0}^{n}\lambda _{i}\sum _{j=0}^{n}\lambda _{i,j}x'_{j}=\sum _{j=0}^{n}{\biggl (}\sum _{i=0}^{n}\lambda _{i}\lambda _{i,j}{\biggr )}x'_{j}\,,}"></dd></dl> giving us coordinates in the second basis as the tuple <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\bigl \{}\sum _{i}\lambda _{i}\lambda _{i,0},\,\dots ,\,{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\bigl \{}\sum _{i}\lambda _{i}\lambda _{i,0},\,\dots ,\,{}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34bab639046458c08c1c0e80b9e8eb638d3cee8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.925ex; height:3.176ex;" alt="{\textstyle {\bigl \{}\sum _{i}\lambda _{i}\lambda _{i,0},\,\dots ,\,{}}"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i}\lambda _{i}\lambda _{i,n}{\bigr \}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i}\lambda _{i}\lambda _{i,n}{\bigr \}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db94f5bb12d3a9c598f585d27c7481f35069c2c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.749ex; height:3.176ex;" alt="{\textstyle \sum _{i}\lambda _{i}\lambda _{i,n}{\bigr \}}}">. <div class="mw-heading mw-heading4"><h4 id="Case_of_affine_coordinates">Case of affine coordinates</h4>[<a href="/w/index.php?title=Affine_space&action=edit&section=21" title="Edit section: Case of affine coordinates">edit</a>]</div> Affine coordinates are also readily changed from one basis to another. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1031f61947aa3d1cf3a70ec3e4904df2c3675d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle o}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{v_{1},\dots ,v_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{v_{1},\dots ,v_{n}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/112d2b712aeabf675d8dbc9a3c02e547fd4a6759" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.031ex; height:2.843ex;" alt="{\displaystyle \{v_{1},\dots ,v_{n}\}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>o</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/865c57d97def04339892a4ce7dc385b428e6d914" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.812ex; height:2.509ex;" alt="{\displaystyle o'}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{v'_{1},\dots ,v'_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{v'_{1},\dots ,v'_{n}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c50561a3278c84f102d11197ea23eb1a8806c15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.031ex; height:3.009ex;" alt="{\displaystyle \{v'_{1},\dots ,v'_{n}\}}"> be affine frames of A. For each point p of A, there is a unique sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1},\dots ,\lambda _{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> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1},\dots ,\lambda _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0980b08cb9c0fae51d9109a1c6319b22807e0eea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.161ex; height:2.509ex;" alt="{\displaystyle \lambda _{1},\dots ,\lambda _{n}}"> of elements of the ground field such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=o+\lambda _{1}v_{1}+\dots +\lambda _{n}v_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>o</mi> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <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> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=o+\lambda _{1}v_{1}+\dots +\lambda _{n}v_{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a241f4da37377a89d62ab447d54cd108ae6abc5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:26.887ex; height:2.509ex;" alt="{\displaystyle p=o+\lambda _{1}v_{1}+\dots +\lambda _{n}v_{n},}"></dd></dl> and similarly, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}\in \{v_{1},\dots ,v_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}\in \{v_{1},\dots ,v_{n}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fedfdca0e248165559f96037a9c4d0093143bd09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.799ex; height:2.843ex;" alt="{\displaystyle v_{i}\in \{v_{1},\dots ,v_{n}\}}"> from the first basis, we now have in the second basis <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o=o'+\lambda _{o,1}v'_{1}+\dots +\lambda _{o,j}v'_{j}+\dots +\lambda _{o,n}v'_{n}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo>=</mo> <msup> <mi>o</mi> <mo>′</mo> </msup> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o=o'+\lambda _{o,1}v'_{1}+\dots +\lambda _{o,j}v'_{j}+\dots +\lambda _{o,n}v'_{n}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e7609d671426ae1c3f033c9db108c478e17600b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:43.652ex; height:3.509ex;" alt="{\displaystyle o=o'+\lambda _{o,1}v'_{1}+\dots +\lambda _{o,j}v'_{j}+\dots +\lambda _{o,n}v'_{n}\,}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}=\lambda _{i,1}v'_{1}+\dots +\lambda _{i,j}v'_{j}+\dots +\lambda _{i,n}v'_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}=\lambda _{i,1}v'_{1}+\dots +\lambda _{i,j}v'_{j}+\dots +\lambda _{i,n}v'_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3af6cfe0b15bd30424c6d33d3e198fdcdac2646f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:38.722ex; height:3.176ex;" alt="{\displaystyle v_{i}=\lambda _{i,1}v'_{1}+\dots +\lambda _{i,j}v'_{j}+\dots +\lambda _{i,n}v'_{n}}"></dd></dl> for tuple <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\lambda _{o,1},\dots ,\lambda _{o,n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\lambda _{o,1},\dots ,\lambda _{o,n}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87ccf42a708bfe172ad8e752d0dd66863921e0f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.996ex; height:3.009ex;" alt="{\displaystyle \{\lambda _{o,1},\dots ,\lambda _{o,n}\}}"> and tuples <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\lambda _{i,1},\dots ,\lambda _{i,n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\lambda _{i,1},\dots ,\lambda _{i,n}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69a04a57d0e79f79458dd96a84ae009c3c590516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.536ex; height:3.009ex;" alt="{\displaystyle \{\lambda _{i,1},\dots ,\lambda _{i,n}\}}">. Now we can rewrite our expression in the first basis as one in the second with <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\,p&=o+\sum _{i=1}^{n}\lambda _{i}v_{i}={\biggl (}o'+\sum _{j=1}^{n}\lambda _{o,j}v'_{j}{\biggr )}+\sum _{i=1}^{n}\lambda _{i}\sum _{j=1}^{n}\lambda _{i,j}v'_{j}\\&=o'+\sum _{j=1}^{n}{\biggl (}\lambda _{o,j}+\sum _{i=1}^{n}\lambda _{i}\lambda _{i,j}{\biggr )}v'_{j}\,,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mspace width="thinmathspace" /> <mi>p</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>o</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <msup> <mi>o</mi> <mo>′</mo> </msup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>o</mi> <mo>′</mo> </msup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <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"> <mi>o</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\,p&=o+\sum _{i=1}^{n}\lambda _{i}v_{i}={\biggl (}o'+\sum _{j=1}^{n}\lambda _{o,j}v'_{j}{\biggr )}+\sum _{i=1}^{n}\lambda _{i}\sum _{j=1}^{n}\lambda _{i,j}v'_{j}\\&=o'+\sum _{j=1}^{n}{\biggl (}\lambda _{o,j}+\sum _{i=1}^{n}\lambda _{i}\lambda _{i,j}{\biggr )}v'_{j}\,,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/655d3c33c028b611572d71d27fecdc3714e379d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:55.864ex; height:14.676ex;" alt="{\displaystyle {\begin{aligned}\,p&=o+\sum _{i=1}^{n}\lambda _{i}v_{i}={\biggl (}o'+\sum _{j=1}^{n}\lambda _{o,j}v'_{j}{\biggr )}+\sum _{i=1}^{n}\lambda _{i}\sum _{j=1}^{n}\lambda _{i,j}v'_{j}\\&=o'+\sum _{j=1}^{n}{\biggl (}\lambda _{o,j}+\sum _{i=1}^{n}\lambda _{i}\lambda _{i,j}{\biggr )}v'_{j}\,,\end{aligned}}}"></dd></dl> giving us coordinates in the second basis as the tuple <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\bigl \{}\lambda _{o,1}+\sum _{i}\lambda _{i}\lambda _{i,1},\,\dots ,\,{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\bigl \{}\lambda _{o,1}+\sum _{i}\lambda _{i}\lambda _{i,1},\,\dots ,\,{}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799a1aa75481a5efc7221fd0d5403f62f6d2364c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.042ex; height:3.176ex;" alt="{\textstyle {\bigl \{}\lambda _{o,1}+\sum _{i}\lambda _{i}\lambda _{i,1},\,\dots ,\,{}}"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lambda _{o,n}+\sum _{i}\lambda _{i}\lambda _{i,n}{\bigr \}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lambda _{o,n}+\sum _{i}\lambda _{i}\lambda _{i,n}{\bigr \}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d3abc0acdecd2471b72cb56216b876b0948f8df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.418ex; height:3.176ex;" alt="{\textstyle \lambda _{o,n}+\sum _{i}\lambda _{i}\lambda _{i,n}{\bigr \}}}">. <div class="mw-heading mw-heading2"><h2 id="Properties_of_affine_homomorphisms">Properties of affine homomorphisms</h2>[<a href="/w/index.php?title=Affine_space&action=edit&section=22" title="Edit section: Properties of affine homomorphisms">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Matrix_representation">Matrix representation</h3>[<a href="/w/index.php?title=Affine_space&action=edit&section=23" title="Edit section: Matrix representation">edit</a>]</div> An affine transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> is executed on a projective space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fb80175bc622b3936c7a0438fc690b2ec410b4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.475ex; height:2.676ex;" alt="{\displaystyle \mathbb {P} ^{3}}"> of <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><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}}">, by a 4 by 4 matrix with a special<a href="#cite_note-8">[8]</a> fourth column: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}a_{11}&a_{12}&a_{13}&0\\a_{21}&a_{22}&a_{23}&0\\a_{31}&a_{32}&a_{33}&0\\a_{41}&a_{42}&a_{43}&1\end{bmatrix}}={\begin{bmatrix}T(1,0,0)&0\\T(0,1,0)&0\\T(0,0,1)&0\\T(0,0,0)&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>41</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>42</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>43</mn> </mrow> </msub> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>T</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>T</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>T</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>T</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}a_{11}&a_{12}&a_{13}&0\\a_{21}&a_{22}&a_{23}&0\\a_{31}&a_{32}&a_{33}&0\\a_{41}&a_{42}&a_{43}&1\end{bmatrix}}={\begin{bmatrix}T(1,0,0)&0\\T(0,1,0)&0\\T(0,0,1)&0\\T(0,0,0)&1\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ba97b2892cb59be096579c1f25f2ea2ecf199a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:45.578ex; height:13.176ex;" alt="{\displaystyle A={\begin{bmatrix}a_{11}&a_{12}&a_{13}&0\\a_{21}&a_{22}&a_{23}&0\\a_{31}&a_{32}&a_{33}&0\\a_{41}&a_{42}&a_{43}&1\end{bmatrix}}={\begin{bmatrix}T(1,0,0)&0\\T(0,1,0)&0\\T(0,0,1)&0\\T(0,0,0)&1\end{bmatrix}}}"> The transformation is affine instead of linear due to the inclusion of point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28c6fd55d5621fd95ca93549660fbb355fd9bd22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (0,0,0)}">, the transformed output of which reveals the affine shift. <div class="mw-heading mw-heading3"><h3 id="Image_and_fibers">Image and fibers</h3>[<a href="/w/index.php?title=Affine_space&action=edit&section=24" title="Edit section: Image and fibers">edit</a>]</div> Let <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon E\to F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon E\to F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65e588635af63181c20287f8ea6f18389b5eaec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.443ex; height:2.509ex;" alt="{\displaystyle f\colon E\to F}"></dd></dl> be an affine homomorphism, with <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>→</mo> </mover> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6627766bc0f5b60d7580e2debdad939ba6445953" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.684ex; height:4.009ex;" alt="{\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}}"></dd></dl> its associated linear map. The image of f is the affine subspace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(E)=\{f(a)\mid a\in E\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>a</mi> <mo>∈</mo> <mi>E</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(E)=\{f(a)\mid a\in E\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16483836fc5c9ce4bfe3d15fdb161a2733e032fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.388ex; height:2.843ex;" alt="{\displaystyle f(E)=\{f(a)\mid a\in E\}}"> of F, which has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {f}}({\overrightarrow {E}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>→</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {f}}({\overrightarrow {E}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1498630d8ac79ab2081477c30721835073e156c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.168ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {f}}({\overrightarrow {E}})}"> as associated vector space. As an affine space does not have a <a href="/wiki/Zero_element" title="Zero element">zero element</a>, an affine homomorphism does not have a <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a>. However, the linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfa88ab873934603aafa809d533b4c49fc16adda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.802ex; height:4.009ex;" alt="{\displaystyle {\overrightarrow {f}}}"> does, and if we denote by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\{v\in {\overrightarrow {E}}\mid {\overrightarrow {f}}(v)=0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>v</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> <mo>∣</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>→</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\{v\in {\overrightarrow {E}}\mid {\overrightarrow {f}}(v)=0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aac122210a6f603ca4e9135006cb555ecd44cae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.951ex; height:4.176ex;" alt="{\displaystyle K=\{v\in {\overrightarrow {E}}\mid {\overrightarrow {f}}(v)=0\}}"> its kernel, then for any point x of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(E)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3465496148d29e6543003de4cdbac8e9ce6beda7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.864ex; height:2.843ex;" alt="{\displaystyle f(E)}">, the <a href="/wiki/Inverse_image" class="mw-redirect" title="Inverse image">inverse image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46a543a3462cbb9effd5d0c514529426172c7d7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.792ex; height:3.176ex;" alt="{\displaystyle f^{-1}(x)}"> of x is an affine subspace of E whose direction is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. This affine subspace is called the <a href="/wiki/Fiber_(mathematics)" title="Fiber (mathematics)">fiber</a> of x. <div class="mw-heading mw-heading3"><h3 id="Projection">Projection</h3>[<a href="/w/index.php?title=Affine_space&action=edit&section=25" title="Edit section: Projection">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Projection_(mathematics)" title="Projection (mathematics)">Projection (mathematics)</a></div> An important example is the projection parallel to some direction onto an affine subspace. The importance of this example lies in the fact that <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean spaces</a> are affine spaces, and that these kinds of projections are fundamental in <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>. More precisely, given an affine space E with associated vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d664d9d5a70d17f42f3b274eb2e79037cf3bcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}}">, let F be an affine subspace of direction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac2b5875f3997034668b7b97ed82ae892697ba9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.677ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {F}}}">, and D be a <a href="/wiki/Complementary_subspace" class="mw-redirect" title="Complementary subspace">complementary subspace</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac2b5875f3997034668b7b97ed82ae892697ba9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.677ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {F}}}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d664d9d5a70d17f42f3b274eb2e79037cf3bcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}}"> (this means that every vector of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d664d9d5a70d17f42f3b274eb2e79037cf3bcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}}"> may be decomposed in a unique way as the sum of an element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac2b5875f3997034668b7b97ed82ae892697ba9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.677ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {F}}}"> and an element of D). For every point x of E, its projection to F parallel to D is the unique point p(x) in F such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)-x\in D.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo>∈</mo> <mi>D</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)-x\in D.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53313df4053989e8d98b5b439ed6e49b0012c9a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.98ex; height:2.843ex;" alt="{\displaystyle p(x)-x\in D.}"></dd></dl> This is an affine homomorphism whose associated linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a48508462cc610fbb2c91ca441740f6b4c5478e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.517ex; height:3.343ex;" alt="{\displaystyle {\overrightarrow {p}}}"> is defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {p}}(x-y)=p(x)-p(y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>→</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {p}}(x-y)=p(x)-p(y),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/916e0b4c9df896231f026b58028771e7ddf9bba8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.68ex; height:3.509ex;" alt="{\displaystyle {\overrightarrow {p}}(x-y)=p(x)-p(y),}"></dd></dl> for x and y in E. The image of this projection is F, and its fibers are the subspaces of direction D. <div class="mw-heading mw-heading3"><h3 id="Quotient_space">Quotient space</h3>[<a href="/w/index.php?title=Affine_space&action=edit&section=26" title="Edit section: Quotient space">edit</a>]</div> Although kernels are not defined for affine spaces, <a href="/wiki/Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">quotient spaces</a> are defined. This results from the fact that "belonging to the same fiber of an affine homomorphism" is an equivalence relation. Let E be an affine space, and D be a <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> of the associated vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d664d9d5a70d17f42f3b274eb2e79037cf3bcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}}">. The quotient E/D of E by D is the <a href="/wiki/Quotient_by_an_equivalence_relation" title="Quotient by an equivalence relation">quotient</a> of E by the <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> such that x and y are equivalent if <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-y\in D.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>∈</mo> <mi>D</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-y\in D.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a75d3c0914bc464c0dfcc8f5705af69b6eb2f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.737ex; height:2.509ex;" alt="{\displaystyle x-y\in D.}"></dd></dl> This quotient is an affine space, which has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {E}}/D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}/D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce5db3f473ca606ea10bbe524def85ed268a0bd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.643ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {E}}/D}"> as associated vector space. For every affine homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\to F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">→</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\to F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84d9e3dcf392a3538226db6bda1b7e2dd73c32e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.13ex; height:2.176ex;" alt="{\displaystyle E\to F}">, the image is isomorphic to the quotient of E by the kernel of the associated linear map. This is the <a href="/wiki/First_isomorphism_theorem" class="mw-redirect" title="First isomorphism theorem">first isomorphism theorem</a> for affine spaces. <div class="mw-heading mw-heading2"><h2 id="Axioms">Axioms</h2>[<a href="/w/index.php?title=Affine_space&action=edit&section=27" title="Edit section: Axioms">edit</a>]</div> Affine spaces are usually studied by <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a> using coordinates, or equivalently vector spaces. They can also be studied as <a href="/wiki/Synthetic_geometry" title="Synthetic geometry">synthetic geometry</a> by writing down axioms, though this approach is much less common. There are several different systems of axioms for affine space. <a href="#CITEREFCoxeter1969">Coxeter (1969</a>, p. 192) axiomatizes the special case of <a href="/wiki/Affine_geometry" title="Affine geometry">affine geometry</a> over the reals as <a href="/wiki/Ordered_geometry" title="Ordered geometry">ordered geometry</a> together with an affine form of <a href="/wiki/Desargues%27s_theorem" title="Desargues's theorem">Desargues's theorem</a> and an axiom stating that in a plane there is at most one line through a given point not meeting a given line. Affine planes satisfy the following axioms (<a href="#CITEREFCameron1991">Cameron 1991</a>, chapter 2): (in which two lines are called parallel if they are equal or disjoint): <ul><li>Any two distinct points lie on a unique line.</li> <li>Given a point and line there is a unique line that contains the point and is parallel to the line</li> <li>There exist three non-collinear points.</li></ul> As well as affine planes over fields (or <a href="/wiki/Division_ring" title="Division ring">division rings</a>), there are also many <a href="/wiki/Non-Desarguesian_plane" title="Non-Desarguesian plane">non-Desarguesian planes</a> satisfying these axioms. (<a href="#CITEREFCameron1991">Cameron 1991</a>, chapter 3) gives axioms for higher-dimensional affine spaces. Purely axiomatic affine geometry is more general than affine spaces and is treated in a <a href="/wiki/Affine_geometry" title="Affine geometry">separate article</a>. <div class="mw-heading mw-heading2"><h2 id="Relation_to_projective_spaces">Relation to projective spaces</h2>[<a href="/w/index.php?title=Affine_space&action=edit&section=28" title="Edit section: Relation to projective spaces">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Affine_space,_projective_space,_vector_space.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Affine_space%2C_projective_space%2C_vector_space.svg/220px-Affine_space%2C_projective_space%2C_vector_space.svg.png" decoding="async" width="220" height="137" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Affine_space%2C_projective_space%2C_vector_space.svg/330px-Affine_space%2C_projective_space%2C_vector_space.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Affine_space%2C_projective_space%2C_vector_space.svg/440px-Affine_space%2C_projective_space%2C_vector_space.svg.png 2x" data-file-width="183" data-file-height="114" /></a><figcaption>An affine space is a subspace of a projective space, which is in turn the quotient of a vector space by an equivalence relation (not by a linear subspace)</figcaption></figure> Affine spaces are contained in <a href="/wiki/Projective_space" title="Projective space">projective spaces</a>. For example, an affine plane can be obtained from any <a href="/wiki/Projective_plane" title="Projective plane">projective plane</a> by removing one line and all the points on it, and conversely any affine plane can be used to construct a projective plane as a <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closure</a> by adding a <a href="/wiki/Line_at_infinity" title="Line at infinity">line at infinity</a> whose points correspond to equivalence classes of <a href="/wiki/Parallel_lines" class="mw-redirect" title="Parallel lines">parallel lines</a>. Similar constructions hold in higher dimensions. Further, transformations of projective space that preserve affine space (equivalently, that leave the <a href="/wiki/Hyperplane_at_infinity" title="Hyperplane at infinity">hyperplane at infinity</a> <a href="/wiki/Invariant_(mathematics)#Invariant_set" title="Invariant (mathematics)">invariant as a set</a>) yield transformations of affine space. Conversely, any affine linear transformation extends uniquely to a <a href="/wiki/Projective_linear_transformation" class="mw-redirect" title="Projective linear transformation">projective linear transformation</a>, so the <a href="/wiki/Affine_group" title="Affine group">affine group</a> is a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of the <a href="/wiki/Projective_group" class="mw-redirect" title="Projective group">projective group</a>. For instance, <a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a> (transformations of the <a href="/wiki/Complex_projective_line" class="mw-redirect" title="Complex projective line">complex projective line</a>, or <a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a>) are affine (transformations of the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>) if and only if they fix the <a href="/wiki/Point_at_infinity" title="Point at infinity">point at infinity</a>. <div class="mw-heading mw-heading2"><h2 id="Affine_algebraic_geometry">Affine algebraic geometry</h2>[<a href="/w/index.php?title=Affine_space&action=edit&section=29" title="Edit section: Affine algebraic geometry">edit</a>]</div> In <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, an <a href="/wiki/Affine_variety" title="Affine variety">affine variety</a> (or, more generally, an <a href="/wiki/Affine_algebraic_set" class="mw-redirect" title="Affine algebraic set">affine algebraic set</a>) is defined as the subset of an affine space that is the set of the common zeros of a set of so-called polynomial functions over the affine space. For defining a polynomial function over the affine space, one has to choose an <a href="/wiki/Affine_frame" class="mw-redirect" title="Affine frame">affine frame</a>. Then, a polynomial function is a function such that the image of any point is the value of some multivariate <a href="/wiki/Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial function</a> of the coordinates of the point. As a change of affine coordinates may be expressed by <a href="/wiki/Linear_function" title="Linear function">linear functions</a> (more precisely affine functions) of the coordinates, this definition is independent of a particular choice of coordinates. The choice of a system of affine coordinates for an affine space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{k}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{k}^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0579559af277736057f7672ec4b75f9027f17dd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.897ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{k}^{n}}"> of dimension n over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> k induces an affine <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{k}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{k}^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0579559af277736057f7672ec4b75f9027f17dd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.897ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{k}^{n}}"> and the affine <a href="/wiki/Coordinate_space" class="mw-redirect" title="Coordinate space">coordinate space</a> kn. This explains why, for simplification, many textbooks write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{k}^{n}=k^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo>=</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{k}^{n}=k^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b160394398931357ea654102a3f7352dd65321cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.425ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{k}^{n}=k^{n}}">, and introduce affine <a href="/wiki/Algebraic_varieties" class="mw-redirect" title="Algebraic varieties">algebraic varieties</a> as the common zeros of polynomial functions over kn.<a href="#cite_note-9">[9]</a> As the whole affine space is the set of the common zeros of the <a href="/wiki/Zero_polynomial" class="mw-redirect" title="Zero polynomial">zero polynomial</a>, affine spaces are affine algebraic varieties. <div class="mw-heading mw-heading3"><h3 id="Ring_of_polynomial_functions">Ring of polynomial functions</h3>[<a href="/w/index.php?title=Affine_space&action=edit&section=30" title="Edit section: Ring of polynomial functions">edit</a>]</div> By the definition above, the choice of an affine frame of an affine space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{k}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{k}^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0579559af277736057f7672ec4b75f9027f17dd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.897ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{k}^{n}}"> allows one to identify the polynomial functions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{k}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{k}^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0579559af277736057f7672ec4b75f9027f17dd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.897ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{k}^{n}}"> with polynomials in n variables, the ith variable representing the function that maps a point to its ith coordinate. It follows that the set of polynomial functions over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{k}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{k}^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0579559af277736057f7672ec4b75f9027f17dd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.897ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{k}^{n}}"> is a <a href="/wiki/Algebra_over_a_field" title="Algebra over a field">k-algebra</a>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\left[\mathbb {A} _{k}^{n}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mrow> <mo>[</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\left[\mathbb {A} _{k}^{n}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc1e918756dde70dfb45492c3b82ae11b349a24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.789ex; height:2.843ex;" alt="{\displaystyle k\left[\mathbb {A} _{k}^{n}\right]}">, which is isomorphic to the <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\left[X_{1},\dots ,X_{n}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mrow> <mo>[</mo> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\left[X_{1},\dots ,X_{n}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41000e80982a7309e905469c64ee71a87ea2c1af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.191ex; height:2.843ex;" alt="{\displaystyle k\left[X_{1},\dots ,X_{n}\right]}">. When one changes coordinates, the isomorphism between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\left[\mathbb {A} _{k}^{n}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mrow> <mo>[</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\left[\mathbb {A} _{k}^{n}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc1e918756dde70dfb45492c3b82ae11b349a24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.789ex; height:2.843ex;" alt="{\displaystyle k\left[\mathbb {A} _{k}^{n}\right]}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[X_{1},\dots ,X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k[X_{1},\dots ,X_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b80417d281d3fa33df1c998af00fcb9a84702804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.804ex; height:2.843ex;" alt="{\displaystyle k[X_{1},\dots ,X_{n}]}"> changes accordingly, and this induces an automorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\left[X_{1},\dots ,X_{n}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mrow> <mo>[</mo> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\left[X_{1},\dots ,X_{n}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41000e80982a7309e905469c64ee71a87ea2c1af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.191ex; height:2.843ex;" alt="{\displaystyle k\left[X_{1},\dots ,X_{n}\right]}">, which maps each indeterminate to a polynomial of degree one. It follows that the <a href="/wiki/Total_degree" class="mw-redirect" title="Total degree">total degree</a> defines a <a href="/wiki/Filtration_(mathematics)" title="Filtration (mathematics)">filtration</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\left[\mathbb {A} _{k}^{n}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mrow> <mo>[</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\left[\mathbb {A} _{k}^{n}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc1e918756dde70dfb45492c3b82ae11b349a24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.789ex; height:2.843ex;" alt="{\displaystyle k\left[\mathbb {A} _{k}^{n}\right]}">, which is independent from the choice of coordinates. The total degree defines also a <a href="/wiki/Graded_ring" title="Graded ring">graduation</a>, but it depends on the choice of coordinates, as a change of affine coordinates may map indeterminates on non-<a href="/wiki/Homogeneous_polynomial" title="Homogeneous polynomial">homogeneous polynomials</a>. <div class="mw-heading mw-heading3"><h3 id="Zariski_topology">Zariski topology</h3>[<a href="/w/index.php?title=Affine_space&action=edit&section=31" title="Edit section: Zariski topology">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Zariski_topology" title="Zariski topology">Zariski topology</a></div> Affine spaces over <a href="/wiki/Topological_field" class="mw-redirect" title="Topological field">topological fields</a>, such as the real or the complex numbers, have a natural <a href="/wiki/Topology_(structure)" class="mw-redirect" title="Topology (structure)">topology</a>. The Zariski topology, which is defined for affine spaces over any field, allows use of topological methods in any case. Zariski topology is the unique topology on an affine space whose <a href="/wiki/Closed_set" title="Closed set">closed sets</a> are <a href="/wiki/Affine_algebraic_set" class="mw-redirect" title="Affine algebraic set">affine algebraic sets</a> (that is sets of the common zeros of polynomial functions over the affine set). As, over a topological field, polynomial functions are continuous, every Zariski closed set is closed for the usual topology, if any. In other words, over a topological field, Zariski topology is <a href="/wiki/Coarser_topology" class="mw-redirect" title="Coarser topology">coarser</a> than the natural topology. There is a natural injective function from an affine space into the set of <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideals</a> (that is the <a href="/wiki/Spectrum_of_a_ring" title="Spectrum of a ring">spectrum</a>) of its ring of polynomial functions. When affine coordinates have been chosen, this function maps the point of coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(a_{1},\dots ,a_{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(a_{1},\dots ,a_{n}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852e439cb0c52c41773d190712883d61ed0b242d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.72ex; height:2.843ex;" alt="{\displaystyle \left(a_{1},\dots ,a_{n}\right)}"> to the <a href="/wiki/Maximal_ideal" title="Maximal ideal">maximal ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle X_{1}-a_{1},\dots ,X_{n}-a_{n}\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle X_{1}-a_{1},\dots ,X_{n}-a_{n}\right\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b143e2a1ac80f23993099727654a4557803e5fff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.522ex; height:2.843ex;" alt="{\displaystyle \left\langle X_{1}-a_{1},\dots ,X_{n}-a_{n}\right\rangle }">. This function is a <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> (for the Zariski topology of the affine space and of the spectrum of the ring of polynomial functions) of the affine space onto the image of the function. The case of an <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed ground field</a> is especially important in algebraic geometry, because, in this case, the homeomorphism above is a map between the affine space and the set of all maximal ideals of the ring of functions (this is <a href="/wiki/Hilbert%27s_Nullstellensatz" title="Hilbert's Nullstellensatz">Hilbert's Nullstellensatz</a>). This is the starting idea of <a href="/wiki/Scheme_theory" class="mw-redirect" title="Scheme theory">scheme theory</a> of <a href="/wiki/Grothendieck" class="mw-redirect" title="Grothendieck">Grothendieck</a>, which consists, for studying algebraic varieties, of considering as "points", not only the points of the affine space, but also all the prime ideals of the spectrum. This allows <a href="/wiki/Gluing_(topology)" class="mw-redirect" title="Gluing (topology)">gluing</a> together algebraic varieties in a similar way as, for <a href="/wiki/Manifold" title="Manifold">manifolds</a>, <a href="/wiki/Chart_(topology)" class="mw-redirect" title="Chart (topology)">charts</a> are glued together for building a manifold. <div class="mw-heading mw-heading3"><h3 id="Cohomology">Cohomology</h3>[<a href="/w/index.php?title=Affine_space&action=edit&section=32" title="Edit section: Cohomology">edit</a>]</div> Like all affine varieties, local data on an affine space can always be patched together globally: the <a href="/wiki/Cohomology" title="Cohomology">cohomology</a> of affine space is trivial. More precisely, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{i}\left(\mathbb {A} _{k}^{n},\mathbf {F} \right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{i}\left(\mathbb {A} _{k}^{n},\mathbf {F} \right)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b7c43b6719672ab32d34c6b3f07f2960b0ffa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.974ex; height:3.176ex;" alt="{\displaystyle H^{i}\left(\mathbb {A} _{k}^{n},\mathbf {F} \right)=0}"> for all <a href="/wiki/Coherent_sheaves" class="mw-redirect" title="Coherent sheaves">coherent sheaves</a> F, and <a href="/wiki/Integer" title="Integer">integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f49f2878fd68a89c3da37eb537198e887cf0293" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle i>0}">. This property is also enjoyed by all other <a href="/wiki/Affine_variety" title="Affine variety">affine varieties</a>. But also all of the <a href="/wiki/%C3%89tale_cohomology" title="Étale cohomology">étale cohomology</a> groups on affine space are trivial. In particular, every <a href="/wiki/Line_bundle" title="Line bundle">line bundle</a> is trivial. More generally, the <a href="/wiki/Quillen%E2%80%93Suslin_theorem" title="Quillen–Suslin theorem">Quillen–Suslin theorem</a> implies that every algebraic <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundle</a> over an affine space is trivial. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Affine_space&action=edit&section=33" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Affine_hull" title="Affine hull">Affine hull</a> – Smallest affine subspace that contains a subset</li> <li><a href="/wiki/Complex_affine_space" title="Complex affine space">Complex affine space</a> – Affine space over the complex numbers</li> <li><a href="/wiki/Dimensional_analysis#Geometry:_position_vs._displacement" title="Dimensional analysis">Dimensional analysis § Geometry: position vs. displacement</a></li> <li><a href="/wiki/Exotic_affine_space" title="Exotic affine space">Exotic affine space</a> – Real affine space of even dimension that is not isomorphic to a complex affine space</li> <li><a href="/wiki/Space_(mathematics)" title="Space (mathematics)">Space (mathematics)</a> – Mathematical set with some added structure</li> <li><a href="/wiki/Barycentric_coordinate_system" title="Barycentric coordinate system">Barycentric coordinate system</a> – Coordinate system that is defined by points instead of vectors</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Affine_space&action=edit&section=34" title="Edit section: Notes">edit</a>]</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-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> The word translation is generally preferred to displacement vector, which may be confusing, as <a href="/wiki/Displacement_(mathematics)" class="mw-redirect" title="Displacement (mathematics)">displacements</a> include also <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotations</a>. </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFBerger1987">Berger 1987</a>, p. 32 </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <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="CITEREFBerger,_Marcel1984" class="citation cs2">Berger, Marcel (1984), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VXRppKJwpaAC&pg=PA11">"Affine spaces"</a>, Problems in Geometry, Springer, p. 11, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387909714" title="Special:BookSources/9780387909714"><bdi>9780387909714</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Affine+spaces&rft.btitle=Problems+in+Geometry&rft.pages=11&rft.pub=Springer&rft.date=1984&rft.isbn=9780387909714&rft.au=Berger%2C+Marcel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVXRppKJwpaAC%26pg%3DPA11&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAffine+space" class="Z3988"> </li> <li id="cite_note-Berger1987-4"><a href="#cite_ref-Berger1987_4-0">^</a> <a href="#CITEREFBerger1987">Berger 1987</a>, p. 33 </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSnapperTroyer1989" class="citation cs2">Snapper, Ernst; Troyer, Robert J. (1989), Metric Affine Geometry, p. 6</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Metric+Affine+Geometry&rft.pages=6&rft.date=1989&rft.aulast=Snapper&rft.aufirst=Ernst&rft.au=Troyer%2C+Robert+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAffine+space" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTarrida,_Agusti_R.2011" class="citation cs2">Tarrida, Agusti R. (2011), "Affine spaces", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UZvxUBzraGAC&pg=PA1">Affine Maps, Euclidean Motions and Quadrics</a>, Springer, pp. 1–2, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780857297105" title="Special:BookSources/9780857297105"><bdi>9780857297105</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Affine+spaces&rft.btitle=Affine+Maps%2C+Euclidean+Motions+and+Quadrics&rft.pages=1-2&rft.pub=Springer&rft.date=2011&rft.isbn=9780857297105&rft.au=Tarrida%2C+Agusti+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DUZvxUBzraGAC%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAffine+space" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFNomizuSasaki1994">Nomizu & Sasaki 1994</a>, p. 7 </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrang2009" class="citation book cs1">Strang, Gilbert (2009). Introduction to Linear Algebra (4th ed.). Wellesley: Wellesley-Cambridge Press. p. 460. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-9802327-1-4" title="Special:BookSources/978-0-9802327-1-4"><bdi>978-0-9802327-1-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Linear+Algebra&rft.place=Wellesley&rft.pages=460&rft.edition=4th&rft.pub=Wellesley-Cambridge+Press&rft.date=2009&rft.isbn=978-0-9802327-1-4&rft.aulast=Strang&rft.aufirst=Gilbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAffine+space" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: date and year (<a href="/wiki/Category:CS1_maint:_date_and_year" title="Category:CS1 maint: date and year">link</a>) </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#CITEREFHartshorne1977">Hartshorne 1977</a>, Ch. I, § 1. </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Affine_space&action=edit&section=35" title="Edit section: References">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerger,_Marcel1984" class="citation cs2">Berger, Marcel (1984), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VXRppKJwpaAC&pg=PA11">"Affine spaces"</a>, Problems in Geometry, Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90971-4" title="Special:BookSources/978-0-387-90971-4"><bdi>978-0-387-90971-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Affine+spaces&rft.btitle=Problems+in+Geometry&rft.pub=Springer-Verlag&rft.date=1984&rft.isbn=978-0-387-90971-4&rft.au=Berger%2C+Marcel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVXRppKJwpaAC%26pg%3DPA11&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAffine+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerger1987" class="citation cs2"><a href="/wiki/Marcel_Berger" title="Marcel Berger">Berger, Marcel</a> (1987), Geometry I, Berlin: Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-11658-3" title="Special:BookSources/3-540-11658-3"><bdi>3-540-11658-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+I&rft.place=Berlin&rft.pub=Springer&rft.date=1987&rft.isbn=3-540-11658-3&rft.aulast=Berger&rft.aufirst=Marcel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAffine+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCameron1991" class="citation cs2"><a href="/wiki/Peter_Cameron_(mathematician)" title="Peter Cameron (mathematician)">Cameron, Peter J.</a> (1991), <a rel="nofollow" class="external text" href="http://www.maths.qmul.ac.uk/~pjc/pps/">Projective and polar spaces</a>, QMW Maths Notes, vol. 13, London: Queen Mary and Westfield College School of Mathematical Sciences, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1153019">1153019</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Projective+and+polar+spaces&rft.place=London&rft.series=QMW+Maths+Notes&rft.pub=Queen+Mary+and+Westfield+College+School+of+Mathematical+Sciences&rft.date=1991&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1153019%23id-name%3DMR&rft.aulast=Cameron&rft.aufirst=Peter+J.&rft_id=http%3A%2F%2Fwww.maths.qmul.ac.uk%2F~pjc%2Fpps%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAffine+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeter1969" class="citation cs2"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, Harold Scott MacDonald</a> (1969), Introduction to Geometry (2nd ed.), New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-50458-0" title="Special:BookSources/978-0-471-50458-0"><bdi>978-0-471-50458-0</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0123930">0123930</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Geometry&rft.place=New+York&rft.edition=2nd&rft.pub=John+Wiley+%26+Sons&rft.date=1969&rft.isbn=978-0-471-50458-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D123930%23id-name%3DMR&rft.aulast=Coxeter&rft.aufirst=Harold+Scott+MacDonald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAffine+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDolgachevShirokov2001" class="citation cs2"><a href="/wiki/Dolgachev" class="mw-redirect" title="Dolgachev">Dolgachev, I.V.</a>; Shirokov, A.P. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Affine_space">"Affine space"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Affine+space&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Dolgachev&rft.aufirst=I.V.&rft.au=Shirokov%2C+A.P.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DAffine_space&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAffine+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartshorne1977" class="citation book cs1"><a href="/wiki/Robin_Hartshorne" title="Robin Hartshorne">Hartshorne, Robin</a> (1977). <a href="/wiki/Algebraic_Geometry_(book)" title="Algebraic Geometry (book)">Algebraic Geometry</a>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90244-9" title="Special:BookSources/978-0-387-90244-9"><bdi>978-0-387-90244-9</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0367.14001">0367.14001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Geometry&rft.pub=Springer-Verlag&rft.date=1977&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0367.14001%23id-name%3DZbl&rft.isbn=978-0-387-90244-9&rft.aulast=Hartshorne&rft.aufirst=Robin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAffine+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNomizuSasaki1994" class="citation cs2"><a href="/wiki/Katsumi_Nomizu" title="Katsumi Nomizu">Nomizu, K.</a>; <a href="/wiki/Shigeo_Sasaki" title="Shigeo Sasaki">Sasaki, S.</a> (1994), <a rel="nofollow" class="external text" href="https://archive.org/details/affinedifferenti0000nomi">Affine Differential Geometry</a> (New ed.), Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-44177-3" title="Special:BookSources/978-0-521-44177-3"><bdi>978-0-521-44177-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Affine+Differential+Geometry&rft.edition=New&rft.pub=Cambridge+University+Press&rft.date=1994&rft.isbn=978-0-521-44177-3&rft.aulast=Nomizu&rft.aufirst=K.&rft.au=Sasaki%2C+S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Faffinedifferenti0000nomi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAffine+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSnapperTroyer1989" class="citation cs2"><a href="/wiki/Ernst_Snapper" title="Ernst Snapper">Snapper, Ernst</a>; Troyer, Robert J. (1989), Metric Affine Geometry (Dover edition, first published in 1989 ed.), Dover Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-66108-3" title="Special:BookSources/0-486-66108-3"><bdi>0-486-66108-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Metric+Affine+Geometry&rft.edition=Dover+edition%2C+first+published+in+1989&rft.pub=Dover+Publications&rft.date=1989&rft.isbn=0-486-66108-3&rft.aulast=Snapper&rft.aufirst=Ernst&rft.au=Troyer%2C+Robert+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAffine+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReventós_Tarrida,_Agustí2011" class="citation cs2">Reventós Tarrida, Agustí (2011), "Affine spaces", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UZvxUBzraGAC&pg=PA1">Affine Maps, Euclidean Motions and Quadrics</a>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-85729-709-9" title="Special:BookSources/978-0-85729-709-9"><bdi>978-0-85729-709-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Affine+spaces&rft.btitle=Affine+Maps%2C+Euclidean+Motions+and+Quadrics&rft.pub=Springer&rft.date=2011&rft.isbn=978-0-85729-709-9&rft.au=Revent%C3%B3s+Tarrida%2C+Agust%C3%AD&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DUZvxUBzraGAC%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAffine+space" class="Z3988"></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><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <a href="https://www.wikidata.org/wiki/Q382698#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></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 rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph118276&CON_LNG=ENG">Czech Republic</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=Affine_space&oldid=1251420333">https://en.wikipedia.org/w/index.php?title=Affine_space&oldid=1251420333</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:Affine_geometry" title="Category:Affine geometry">Affine geometry</a></li><li><a href="/wiki/Category:Linear_algebra" title="Category:Linear algebra">Linear algebra</a></li><li><a href="/wiki/Category:Space_(mathematics)" title="Category:Space (mathematics)">Space (mathematics)</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:CS1_maint:_date_and_year" title="Category:CS1 maint: date and year">CS1 maint: date and year</a></li><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 16 October 2024, at 01:30 (UTC).</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=Affine_space&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-694cf4987f-h5dm9","wgBackendResponseTime":193,"wgPageParseReport":{"limitreport":{"cputime":"0.854","walltime":"1.047","ppvisitednodes":{"value":12017,"limit":1000000},"postexpandincludesize":{"value":65072,"limit":2097152},"templateargumentsize":{"value":15060,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":7,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":52849,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 732.487 1 -total"," 23.70% 173.571 185 Template:Math"," 18.63% 136.445 5 Template:Annotated_link"," 15.66% 114.686 1 Template:Reflist"," 15.65% 114.645 10 Template:Citation"," 10.00% 73.246 1 Template:Authority_control"," 9.57% 70.085 1 Template:Short_description"," 6.79% 49.722 2 Template:Pagetype"," 5.26% 38.538 1 Template:Harvtxt"," 5.08% 37.246 190 Template:Main_other"]},"scribunto":{"limitreport-timeusage":{"value":"0.439","limit":"10.000"},"limitreport-memusage":{"value":18850981,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFBerger,_Marcel1984\"] = 2,\n [\"CITEREFBerger1987\"] = 1,\n [\"CITEREFCameron1991\"] = 1,\n [\"CITEREFCoxeter1969\"] = 1,\n [\"CITEREFHartshorne1977\"] = 1,\n [\"CITEREFNomizuSasaki1994\"] = 1,\n [\"CITEREFReventós_Tarrida,_Agustí2011\"] = 1,\n [\"CITEREFSnapperTroyer1989\"] = 2,\n [\"CITEREFStrang2009\"] = 1,\n [\"CITEREFTarrida,_Agusti_R.2011\"] = 1,\n}\ntemplate_list = table#1 {\n [\"=\"] = 1,\n [\"Annotated link\"] = 5,\n [\"Authority control\"] = 1,\n [\"Citation\"] = 10,\n [\"Cite book\"] = 2,\n [\"DEFAULTSORT:Affine Space\"] = 1,\n [\"Distinguish\"] = 1,\n [\"Eom\"] = 1,\n [\"Harv\"] = 2,\n [\"Harvard citation no brackets\"] = 3,\n [\"Harvnb\"] = 1,\n [\"Harvtxt\"] = 1,\n [\"Main\"] = 1,\n [\"Math\"] = 185,\n [\"Mvar\"] = 20,\n [\"Reflist\"] = 1,\n [\"Section link\"] = 1,\n [\"See also\"] = 3,\n [\"Sfrac\"] = 3,\n [\"Short description\"] = 1,\n [\"Slink\"] = 2,\n [\"Vanchor\"] = 1,\n [\"Visible anchor\"] = 2,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-api-ext.codfw.main-744c7589dd-j9dtp","timestamp":"20241125143911","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Affine space","url":"https:\/\/en.wikipedia.org\/wiki\/Affine_space","sameAs":"http:\/\/www.wikidata.org\/entity\/Q382698","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q382698","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":"2003-08-18T04:32:19Z","dateModified":"2024-10-16T01:30:05Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/9\/95\/Affine_space_R3.png","headline":"geometric structure that generalizes the Euclidean space"}</script> </body> </html>

CINXE.COM

Affine space - Wikipedia