Linear combination

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<span class="vector-toc-numb">2</span> <span>Examples and counterexamples</span> </div> </a> <button aria-controls="toc-Examples_and_counterexamples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples and counterexamples subsection</span> </button> <ul id="toc-Examples_and_counterexamples-sublist" class="vector-toc-list"> <li id="toc-Euclidean_vectors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Euclidean_vectors"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Euclidean vectors</span> </div> </a> <ul id="toc-Euclidean_vectors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Functions</span> </div> </a> <ul id="toc-Functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polynomials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polynomials"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Polynomials</span> </div> </a> <ul id="toc-Polynomials-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-The_linear_span" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_linear_span"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>The linear span</span> </div> </a> <ul id="toc-The_linear_span-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linear_independence" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Linear_independence"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Linear independence</span> </div> </a> <ul id="toc-Linear_independence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Affine,_conical,_and_convex_combinations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Affine,_conical,_and_convex_combinations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Affine, conical, and convex combinations</span> </div> </a> <ul id="toc-Affine,_conical,_and_convex_combinations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Operad_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Operad_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Operad theory</span> </div> </a> <ul id="toc-Operad_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Generalizations</span> </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Textbook" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Textbook"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Textbook</span> </div> </a> <ul id="toc-Textbook-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Web" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Web"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>Web</span> </div> </a> <ul id="toc-Web-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Linear combination</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 36 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-36" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">36 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%B1%D9%83%D9%8A%D8%A8_%D8%AE%D8%B7%D9%8A" title="تركيب خطي – Arabic" lang="ar" hreflang="ar" data-title="تركيب خطي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Combinaci%C3%B3_lineal" title="Combinació lineal – Catalan" lang="ca" hreflang="ca" data-title="Combinació lineal" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B8%D0%BB%D0%BB%D0%B5_%D0%BA%D0%BE%D0%BC%D0%B1%D0%B8%D0%BD%D0%B0%D1%86%D0%B8" title="Линилле комбинаци – Chuvash" lang="cv" hreflang="cv" data-title="Линилле комбинаци" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Line%C3%A1rn%C3%AD_kombinace" title="Lineární kombinace – Czech" lang="cs" hreflang="cs" data-title="Lineární kombinace" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Linearkombination" title="Linearkombination – German" lang="de" hreflang="de" data-title="Linearkombination" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Combinaci%C3%B3n_lineal" title="Combinación lineal – Spanish" lang="es" hreflang="es" data-title="Combinación lineal" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Lineara_kombina%C4%B5o" title="Lineara kombinaĵo – Esperanto" lang="eo" hreflang="eo" data-title="Lineara kombinaĵo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%B1%DA%A9%DB%8C%D8%A8_%D8%AE%D8%B7%DB%8C" title="ترکیب خطی – Persian" lang="fa" hreflang="fa" data-title="ترکیب خطی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Combinaison_lin%C3%A9aire" title="Combinaison linéaire – French" lang="fr" hreflang="fr" data-title="Combinaison linéaire" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Combinaci%C3%B3n_linear" title="Combinación linear – Galician" lang="gl" hreflang="gl" data-title="Combinación linear" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%84%A0%ED%98%95_%EA%B2%B0%ED%95%A9" title="선형 결합 – Korean" lang="ko" hreflang="ko" data-title="선형 결합" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Linearna_kombinacija" title="Linearna kombinacija – Croatian" lang="hr" hreflang="hr" data-title="Linearna kombinacija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kombinasi_linear" title="Kombinasi linear – Indonesian" lang="id" hreflang="id" data-title="Kombinasi linear" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Combinazione_lineare" title="Combinazione lineare – Italian" lang="it" hreflang="it" data-title="Combinazione lineare" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A6%D7%99%D7%A8%D7%95%D7%A3_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99" title="צירוף ליניארי – Hebrew" lang="he" hreflang="he" data-title="צירוף ליניארי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Line%C3%A1ris_kombin%C3%A1ci%C3%B3" title="Lineáris kombináció – Hungarian" lang="hu" hreflang="hu" data-title="Lineáris kombináció" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B0%E0%B5%87%E0%B4%96%E0%B5%80%E0%B4%AF%E0%B4%B8%E0%B4%9E%E0%B5%8D%E0%B4%9A%E0%B4%AF%E0%B4%82" title="രേഖീയസഞ്ചയം – Malayalam" lang="ml" hreflang="ml" data-title="രേഖീയസഞ്ചയം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lineaire_combinatie" title="Lineaire combinatie – Dutch" lang="nl" hreflang="nl" data-title="Lineaire combinatie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B7%9A%E5%9E%8B%E7%B5%90%E5%90%88" title="線型結合 – Japanese" lang="ja" hreflang="ja" data-title="線型結合" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Line%C3%A6rkombinasjon" title="Lineærkombinasjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Lineærkombinasjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Kombinacja_liniowa" title="Kombinacja liniowa – Polish" lang="pl" hreflang="pl" data-title="Kombinacja liniowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Combina%C3%A7%C3%A3o_linear" title="Combinação linear – Portuguese" lang="pt" hreflang="pt" data-title="Combinação linear" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%B0%D1%8F_%D0%BA%D0%BE%D0%BC%D0%B1%D0%B8%D0%BD%D0%B0%D1%86%D0%B8%D1%8F" title="Линейная комбинация – Russian" lang="ru" hreflang="ru" data-title="Линейная комбинация" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Linear_combination" title="Linear combination – Simple English" lang="en-simple" hreflang="en-simple" data-title="Linear combination" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Line%C3%A1rna_kombin%C3%A1cia" title="Lineárna kombinácia – Slovak" lang="sk" hreflang="sk" data-title="Lineárna kombinácia" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Linearna_kombinacija" title="Linearna kombinacija – Slovenian" lang="sl" hreflang="sl" data-title="Linearna kombinacija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Lineaarikombinaatio" title="Lineaarikombinaatio – Finnish" lang="fi" hreflang="fi" data-title="Lineaarikombinaatio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Linj%C3%A4rkombination" title="Linjärkombination – Swedish" lang="sv" hreflang="sv" data-title="Linjärkombination" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AF%87%E0%AE%B0%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%9A%E0%AF%87%E0%AE%B0%E0%AF%8D%E0%AE%B5%E0%AF%81" title="நேரியல் சேர்வு – Tamil" lang="ta" hreflang="ta" data-title="நேரியல் சேர்வு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9C%E0%B8%A5%E0%B8%A3%E0%B8%A7%E0%B8%A1%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B9%80%E0%B8%AA%E0%B9%89%E0%B8%99" title="ผลรวมเชิงเส้น – Thai" lang="th" hreflang="th" data-title="ผลรวมเชิงเส้น" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Do%C4%9Frusal_birle%C5%9Fim" title="Doğrusal birleşim – Turkish" lang="tr" hreflang="tr" data-title="Doğrusal birleşim" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9B%D1%96%D0%BD%D1%96%D0%B9%D0%BD%D0%B0_%D0%BA%D0%BE%D0%BC%D0%B1%D1%96%D0%BD%D0%B0%D1%86%D1%96%D1%8F" title="Лінійна комбінація – Ukrainian" lang="uk" hreflang="uk" data-title="Лінійна комбінація" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AE%D8%B7%DB%8C_%D8%A7%D8%AC%D8%AA%D9%85%D8%A7%D8%B9" title="خطی اجتماع – Urdu" lang="ur" hreflang="ur" data-title="خطی اجتماع" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%E1%BB%95_h%E1%BB%A3p_tuy%E1%BA%BFn_t%C3%ADnh" title="Tổ hợp tuyến tính – Vietnamese" lang="vi" hreflang="vi" data-title="Tổ hợp tuyến tính" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%B7%9A%E6%80%A7%E7%B5%84%E5%90%88" title="線性組合 – Cantonese" lang="yue" hreflang="yue" data-title="線性組合" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%BA%BF%E6%80%A7%E7%BB%84%E5%90%88" title="线性组合 – Chinese" lang="zh" hreflang="zh" data-title="线性组合" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q27628#sitelinks-wikipedia" title="Edit interlanguage links" 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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">"Superposition" redirects here. For other uses, see <a href="/wiki/Superposition_(disambiguation)" class="mw-disambig" title="Superposition (disambiguation)">superposition (disambiguation)</a>.</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>linear combination</b> or <b>superposition</b> is an <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">expression</a> constructed from a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of <i>x</i> and <i>y</i> would be any expression of the form <i>ax</i> + <i>by</i>, where <i>a</i> and <i>b</i> are constants).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The concept of linear combinations is central to <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> and related fields of mathematics. Most of this article deals with linear combinations in the context of a <a href="/wiki/Vector_space" title="Vector space">vector space</a> over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, with some generalizations given at the end of the article. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_combination&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>V</i> be a <a href="/wiki/Vector_space" title="Vector space">vector space</a> over the field <i>K</i>. As usual, we call elements of <i>V</i> <i><a href="/wiki/Vector_space" title="Vector space">vectors</a></i> and call elements of <i>K</i> <i><a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a></i>. If <b>v</b><sub>1</sub>,...,<b>v</b><sub><i>n</i></sub> are vectors and <i>a</i><sub>1</sub>,...,<i>a</i><sub><i>n</i></sub> are scalars, then the <i>linear combination of those vectors with those scalars as coefficients</i> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+a_{3}\mathbf {v} _{3}+\cdots +a_{n}\mathbf {v} _{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+a_{3}\mathbf {v} _{3}+\cdots +a_{n}\mathbf {v} _{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06a8dab6b20c2d8f396fac1847e39cc3e1a3e7dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.057ex; height:2.343ex;" alt="{\displaystyle a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+a_{3}\mathbf {v} _{3}+\cdots +a_{n}\mathbf {v} _{n}.}"></span></dd></dl> <p>There is some ambiguity in the use of the term "linear combination" as to whether it refers to the expression or to its value. In most cases the value is emphasized, as in the assertion "the set of all linear combinations of <b>v</b><sub>1</sub>,...,<b>v</b><sub><i>n</i></sub> always forms a subspace". However, one could also say "two different linear combinations can have the same value" in which case the reference is to the expression. The subtle difference between these uses is the essence of the notion of <a href="/wiki/Linear_dependence" class="mw-redirect" title="Linear dependence">linear dependence</a>: a family <i>F</i> of vectors is linearly independent precisely if any linear combination of the vectors in <i>F</i> (as value) is uniquely so (as expression). In any case, even when viewed as expressions, all that matters about a linear combination is the coefficient of each <b>v</b><sub><i>i</i></sub>; trivial modifications such as permuting the terms or adding terms with zero coefficient do not produce distinct linear combinations. </p><p>In a given situation, <i>K</i> and <i>V</i> may be specified explicitly, or they may be obvious from context. In that case, we often speak of <i>a linear combination of the vectors</i> <b>v</b><sub>1</sub>,...,<b>v</b><sub><i>n</i></sub>, with the coefficients unspecified (except that they must belong to <i>K</i>). Or, if <i>S</i> is a <a href="/wiki/Subset" title="Subset">subset</a> of <i>V</i>, we may speak of <i>a linear combination of vectors in S</i>, where both the coefficients and the vectors are unspecified, except that the vectors must belong to the set <i>S</i> (and the coefficients must belong to <i>K</i>). Finally, we may speak simply of <i>a linear combination</i>, where nothing is specified (except that the vectors must belong to <i>V</i> and the coefficients must belong to <i>K</i>); in this case one is probably referring to the expression, since every vector in <i>V</i> is certainly the value of some linear combination. </p><p>Note that by definition, a linear combination involves only <a href="/wiki/Finite_set" title="Finite set">finitely</a> many vectors (except as described in the <a href="#Generalizations">§ Generalizations</a> section. However, the set <i>S</i> that the vectors are taken from (if one is mentioned) can still be <a href="/wiki/Infinity" title="Infinity">infinite</a>; each individual linear combination will only involve finitely many vectors. Also, there is no reason that <i>n</i> cannot be <a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">zero</a>; in that case, we declare by convention that the result of the linear combination is the <a href="/wiki/Zero_vector" class="mw-redirect" title="Zero vector">zero vector</a> in <i>V</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples_and_counterexamples">Examples and counterexamples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_combination&action=edit&section=2" title="Edit section: Examples and counterexamples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-No_footnotes plainlinks metadata ambox ambox-style ambox-No_footnotes" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section includes a <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">list of references</a>, <a href="/wiki/Wikipedia:Further_reading" title="Wikipedia:Further reading">related reading</a>, or <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a>, <b>but its sources remain unclear because it lacks <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this section by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">August 2013</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Euclidean_vectors">Euclidean vectors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_combination&action=edit&section=3" title="Edit section: Euclidean vectors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let the field <i>K</i> be the set <b>R</b> of <a href="/wiki/Real_number" title="Real number">real numbers</a>, and let the vector space <i>V</i> be the <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> <b>R</b><sup>3</sup>. Consider the vectors <span class="nowrap"><b>e</b><sub>1</sub> = (1,0,0)</span>, <span class="nowrap"><b>e</b><sub>2</sub> = (0,1,0)</span> and <span class="nowrap"><b>e</b><sub>3</sub> = (0,0,1)</span>. Then <i>any</i> <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> in <b>R</b><sup>3</sup> is a linear combination of <b>e</b><sub>1</sub>, <b>e</b><sub>2</sub>, and <b>e</b><sub>3</sub>. </p><p>To see that this is so, take an arbitrary vector (<i>a</i><sub>1</sub>,<i>a</i><sub>2</sub>,<i>a</i><sub>3</sub>) in <b>R</b><sup>3</sup>, and write: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(a_{1},a_{2},a_{3})&=(a_{1},0,0)+(0,a_{2},0)+(0,0,a_{3})\\[6pt]&=a_{1}(1,0,0)+a_{2}(0,1,0)+a_{3}(0,0,1)\\[6pt]&=a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3}.\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="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(a_{1},a_{2},a_{3})&=(a_{1},0,0)+(0,a_{2},0)+(0,0,a_{3})\\[6pt]&=a_{1}(1,0,0)+a_{2}(0,1,0)+a_{3}(0,0,1)\\[6pt]&=a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe009e7f4298edf1f8c3cb4a91ad37cf55138972" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:49.206ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}(a_{1},a_{2},a_{3})&=(a_{1},0,0)+(0,a_{2},0)+(0,0,a_{3})\\[6pt]&=a_{1}(1,0,0)+a_{2}(0,1,0)+a_{3}(0,0,1)\\[6pt]&=a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3}.\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Functions">Functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_combination&action=edit&section=4" title="Edit section: Functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>K</i> be the set <b>C</b> of all <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, and let <i>V</i> be the set C<sub><b>C</b></sub>(<i>R</i>) of all <a href="/wiki/Continuous_function" title="Continuous function">continuous functions</a> from the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a> <b>R</b> to the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> <b>C</b>. Consider the vectors (functions) <i>f</i> and <i>g</i> defined by <i>f</i>(<i>t</i>) := <i>e</i><sup><i>it</i></sup> and <i>g</i>(<i>t</i>) := <i>e</i><sup>−<i>it</i></sup>. (Here, <i>e</i> is the <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">base of the natural logarithm</a>, about 2.71828..., and <i>i</i> is the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>, a square root of −1.) Some linear combinations of <i>f</i> and <i>g</i> are: </p> <ul><li><div style="vertical-align: 0%;display:inline;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos t={\tfrac {1}{2}}\,e^{it}+{\tfrac {1}{2}}\,e^{-it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos t={\tfrac {1}{2}}\,e^{it}+{\tfrac {1}{2}}\,e^{-it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9caf0dd7272872ad9b6048360604e016136033d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.599ex; height:3.509ex;" alt="{\displaystyle \cos t={\tfrac {1}{2}}\,e^{it}+{\tfrac {1}{2}}\,e^{-it}}"></span></div></li> <li><div style="vertical-align: 0%;display:inline;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\sin t=(-i)e^{it}+(i)e^{-it}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>t</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>t</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\sin t=(-i)e^{it}+(i)e^{-it}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a9ce545de87b3da60ed2d598aeb2c731f748ce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.482ex; height:3.176ex;" alt="{\displaystyle 2\sin t=(-i)e^{it}+(i)e^{-it}.}"></span></div></li></ul> <p>On the other hand, the constant function 3 is <i>not</i> a linear combination of <i>f</i> and <i>g</i>. To see this, suppose that 3 could be written as a linear combination of <i>e</i><sup><i>it</i></sup> and <i>e</i><sup>−<i>it</i></sup>. This means that there would exist complex scalars <i>a</i> and <i>b</i> such that <span class="nowrap"><i>ae</i><sup><i>it</i></sup> + <i>be</i><sup>−<i>it</i></sup> = 3</span> for all real numbers <i>t</i>. Setting <i>t</i> = 0 and <i>t</i> = π gives the equations <span class="nowrap"><i>a</i> + <i>b</i> = 3</span> and <span class="nowrap"><i>a</i> + <i>b</i> = −3</span>, and clearly this cannot happen. See <a href="/wiki/Euler%27s_identity" title="Euler's identity">Euler's identity</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Polynomials">Polynomials</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_combination&action=edit&section=5" title="Edit section: Polynomials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>K</i> be <b>R</b>, <b>C</b>, or any field, and let <i>V</i> be the set <i>P</i> of all <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> with coefficients taken from the field <i>K</i>. Consider the vectors (polynomials) <i>p</i><sub>1</sub> := 1, <span class="nowrap"><i>p</i><sub>2</sub> := <i>x</i> + 1</span>, and <span class="nowrap"><i>p</i><sub>3</sub> := <i>x</i><sup>2</sup> + <i>x</i> + 1</span>. </p><p>Is the polynomial <i>x</i><sup>2</sup> − 1 a linear combination of <i>p</i><sub>1</sub>, <i>p</i><sub>2</sub>, and <i>p</i><sub>3</sub>? To find out, consider an arbitrary linear combination of these vectors and try to see when it equals the desired vector <i>x</i><sup>2</sup> − 1. Picking arbitrary coefficients <i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, and <i>a</i><sub>3</sub>, we want </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}(1)+a_{2}(x+1)+a_{3}(x^{2}+x+1)=x^{2}-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}(1)+a_{2}(x+1)+a_{3}(x^{2}+x+1)=x^{2}-1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6115ddcd36c0c6733807246de416d226a2a3f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.145ex; height:3.176ex;" alt="{\displaystyle a_{1}(1)+a_{2}(x+1)+a_{3}(x^{2}+x+1)=x^{2}-1.}"></span></dd></dl> <p>Multiplying the polynomials out, this means </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1})+(a_{2}x+a_{2})+(a_{3}x^{2}+a_{3}x+a_{3})=x^{2}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1})+(a_{2}x+a_{2})+(a_{3}x^{2}+a_{3}x+a_{3})=x^{2}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cfebc344eb126e2f4bc4cc3874512758d4bfa0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.863ex; height:3.176ex;" alt="{\displaystyle (a_{1})+(a_{2}x+a_{2})+(a_{3}x^{2}+a_{3}x+a_{3})=x^{2}-1}"></span></dd></dl> <p>and collecting like powers of <i>x</i>, we get </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{3}x^{2}+(a_{2}+a_{3})x+(a_{1}+a_{2}+a_{3})=1x^{2}+0x+(-1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>0</mn> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{3}x^{2}+(a_{2}+a_{3})x+(a_{1}+a_{2}+a_{3})=1x^{2}+0x+(-1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e947732aab5d7355541468b439980019357ee87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.483ex; height:3.176ex;" alt="{\displaystyle a_{3}x^{2}+(a_{2}+a_{3})x+(a_{1}+a_{2}+a_{3})=1x^{2}+0x+(-1).}"></span></dd></dl> <p>Two polynomials are equal <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> their corresponding coefficients are equal, so we can conclude </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{3}=1,\quad a_{2}+a_{3}=0,\quad a_{1}+a_{2}+a_{3}=-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{3}=1,\quad a_{2}+a_{3}=0,\quad a_{1}+a_{2}+a_{3}=-1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa859cbabc345c7c2b1f1029f5a68957ae03e7cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:44.176ex; height:2.509ex;" alt="{\displaystyle a_{3}=1,\quad a_{2}+a_{3}=0,\quad a_{1}+a_{2}+a_{3}=-1.}"></span></dd></dl> <p>This <a href="/wiki/System_of_linear_equations" title="System of linear equations">system of linear equations</a> can easily be solved. First, the first equation simply says that <i>a</i><sub>3</sub> is 1. Knowing that, we can solve the second equation for <i>a</i><sub>2</sub>, which comes out to −1. Finally, the last equation tells us that <i>a</i><sub>1</sub> is also −1. Therefore, the only possible way to get a linear combination is with these coefficients. Indeed, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}-1=-1-(x+1)+(x^{2}+x+1)=-p_{1}-p_{2}+p_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-1=-1-(x+1)+(x^{2}+x+1)=-p_{1}-p_{2}+p_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42bc7f8e4e0e20316742991f0716610c916302b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.903ex; height:3.176ex;" alt="{\displaystyle x^{2}-1=-1-(x+1)+(x^{2}+x+1)=-p_{1}-p_{2}+p_{3}}"></span></dd></dl> <p>so <i>x</i><sup>2</sup> − 1 <i>is</i> a linear combination of <i>p</i><sub>1</sub>, <i>p</i><sub>2</sub>, and <i>p</i><sub>3</sub>. </p><p>On the other hand, what about the polynomial <i>x</i><sup>3</sup> − 1? If we try to make this vector a linear combination of <i>p</i><sub>1</sub>, <i>p</i><sub>2</sub>, and <i>p</i><sub>3</sub>, then following the same process as before, we get the equation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&0x^{3}+a_{3}x^{2}+(a_{2}+a_{3})x+(a_{1}+a_{2}+a_{3})\\[5pt]={}&1x^{3}+0x^{2}+0x+(-1).\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="0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mn>0</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mn>1</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>0</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>0</mn> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&0x^{3}+a_{3}x^{2}+(a_{2}+a_{3})x+(a_{1}+a_{2}+a_{3})\\[5pt]={}&1x^{3}+0x^{2}+0x+(-1).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afb000b9840823f66f23ced29ab3970f598eba9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.83ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}&0x^{3}+a_{3}x^{2}+(a_{2}+a_{3})x+(a_{1}+a_{2}+a_{3})\\[5pt]={}&1x^{3}+0x^{2}+0x+(-1).\end{aligned}}}"></span></dd></dl> <p>However, when we set corresponding coefficients equal in this case, the equation for <i>x</i><sup>3</sup> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2bb1a7bcdbd7e274b7b8581e4357f09dbeee7fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.423ex; height:2.176ex;" alt="{\displaystyle 0=1}"></span></dd></dl> <p>which is always false. Therefore, there is no way for this to work, and <i>x</i><sup>3</sup> − 1 is <i>not</i> a linear combination of <i>p</i><sub>1</sub>, <i>p</i><sub>2</sub>, and <i>p</i><sub>3</sub>. </p> <div class="mw-heading mw-heading2"><h2 id="The_linear_span">The linear span</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_combination&action=edit&section=6" title="Edit section: The linear span"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Linear_span" title="Linear span">Linear span</a></div> <p>Take an arbitrary field <i>K</i>, an arbitrary vector space <i>V</i>, and let <b>v</b><sub>1</sub>,...,<b>v</b><sub><i>n</i></sub> be vectors (in <i>V</i>). It is interesting to consider the set of <i>all</i> linear combinations of these vectors. This set is called the <i><a href="/wiki/Linear_span" title="Linear span">linear span</a></i> (or just <i>span</i>) of the vectors, say <i>S</i> = {<b>v</b><sub>1</sub>, ..., <b>v</b><sub><i>n</i></sub>}. We write the span of <i>S</i> as span(<i>S</i>)<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> or sp(<i>S</i>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {span} (\mathbf {v} _{1},\ldots ,\mathbf {v} _{n}):=\{a_{1}\mathbf {v} _{1}+\cdots +a_{n}\mathbf {v} _{n}:a_{1},\ldots ,a_{n}\in K\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>span</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>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> <mo>∈</mo> <mi>K</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {span} (\mathbf {v} _{1},\ldots ,\mathbf {v} _{n}):=\{a_{1}\mathbf {v} _{1}+\cdots +a_{n}\mathbf {v} _{n}:a_{1},\ldots ,a_{n}\in K\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef16da42a979f6f6f816ad6272d17b1c967775f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.448ex; height:2.843ex;" alt="{\displaystyle \operatorname {span} (\mathbf {v} _{1},\ldots ,\mathbf {v} _{n}):=\{a_{1}\mathbf {v} _{1}+\cdots +a_{n}\mathbf {v} _{n}:a_{1},\ldots ,a_{n}\in K\}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Linear_independence">Linear independence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_combination&action=edit&section=7" title="Edit section: Linear independence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Linear_independence" title="Linear independence">Linear independence</a></div> <p>Suppose that, for some sets of vectors <b>v</b><sub>1</sub>,...,<b>v</b><sub><i>n</i></sub>, a single vector can be written in two different ways as a linear combination of them: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =\sum _{i}a_{i}\mathbf {v} _{i}=\sum _{i}b_{i}\mathbf {v} _{i}{\text{ where }}a_{i}\neq b_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> where </mtext> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≠</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =\sum _{i}a_{i}\mathbf {v} _{i}=\sum _{i}b_{i}\mathbf {v} _{i}{\text{ where }}a_{i}\neq b_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6b1fa5f144d43a3756d0a062fa4611f2457f7ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.02ex; height:5.509ex;" alt="{\displaystyle \mathbf {v} =\sum _{i}a_{i}\mathbf {v} _{i}=\sum _{i}b_{i}\mathbf {v} _{i}{\text{ where }}a_{i}\neq b_{i}.}"></span></dd></dl> <p>This is equivalent, by subtracting these (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}:=a_{i}-b_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i}:=a_{i}-b_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4d3008ec40ae4edaa8eacf9d546d595753d7a10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.219ex; height:2.509ex;" alt="{\displaystyle c_{i}:=a_{i}-b_{i}}"></span>), to saying a non-trivial combination is zero:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} =\sum _{i}c_{i}\mathbf {v} _{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} =\sum _{i}c_{i}\mathbf {v} _{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22e3abb22b48074ec991013f1800319118477119" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.841ex; height:5.509ex;" alt="{\displaystyle \mathbf {0} =\sum _{i}c_{i}\mathbf {v} _{i}.}"></span></dd></dl> <p>If that is possible, then <b>v</b><sub>1</sub>,...,<b>v</b><sub><i>n</i></sub> are called <i><a href="/wiki/Linearly_dependent" class="mw-redirect" title="Linearly dependent">linearly dependent</a></i>; otherwise, they are <i>linearly independent</i>. Similarly, we can speak of linear dependence or independence of an arbitrary set <i>S</i> of vectors. </p><p>If <i>S</i> is linearly independent and the span of <i>S</i> equals <i>V</i>, then <i>S</i> is a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> for <i>V</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Affine,_conical,_and_convex_combinations"><span id="Affine.2C_conical.2C_and_convex_combinations"></span>Affine, conical, and convex combinations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_combination&action=edit&section=8" title="Edit section: Affine, conical, and convex combinations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By restricting the coefficients used in linear combinations, one can define the related concepts of <a href="/wiki/Affine_combination" title="Affine combination">affine combination</a>, <a href="/wiki/Conical_combination" title="Conical combination">conical combination</a>, and <a href="/wiki/Convex_combination" title="Convex combination">convex combination</a>, and the associated notions of sets closed under these operations. </p> <table class="wikitable" style="text-align: left;"> <tbody><tr> <th>Type of combination</th> <th>Restrictions on coefficients</th> <th>Name of set</th> <th>Model space </th></tr> <tr> <td>Linear combination</td> <td>no restrictions</td> <td><a href="/wiki/Vector_subspace" class="mw-redirect" title="Vector subspace">Vector subspace</a></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed657d7f0d7aa156e7b9b171f22b4a3aa6482c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.222ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{n}}"></span> </td></tr> <tr> <td><a href="/wiki/Affine_combination" title="Affine combination">Affine combination</a></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum a_{i}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum a_{i}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/127931a1f73db310d9661793631ab0ec58400dc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.131ex; height:2.843ex;" alt="{\textstyle \sum a_{i}=1}"></span></td> <td><a href="/wiki/Affine_subspace" class="mw-redirect" title="Affine subspace">Affine subspace</a></td> <td>Affine <a href="/wiki/Hyperplane" title="Hyperplane">hyperplane</a> </td></tr> <tr> <td><a href="/wiki/Conical_combination" title="Conical combination">Conical combination</a></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}\geq 0}"> <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> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aefe5d34ff87d51c5218190cb4139ec9e35d8a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.29ex; height:2.509ex;" alt="{\displaystyle a_{i}\geq 0}"></span></td> <td><a href="/wiki/Convex_cone" title="Convex cone">Convex cone</a></td> <td><a href="/wiki/Quadrant_(plane_geometry)" title="Quadrant (plane geometry)">Quadrant</a>, <a href="/wiki/Octant_(solid_geometry)" title="Octant (solid geometry)">octant</a>, or <a href="/wiki/Orthant" title="Orthant">orthant</a> </td></tr> <tr> <td><a href="/wiki/Convex_combination" title="Convex combination">Convex combination</a></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}\geq 0}"> <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> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aefe5d34ff87d51c5218190cb4139ec9e35d8a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.29ex; height:2.509ex;" alt="{\displaystyle a_{i}\geq 0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum a_{i}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum a_{i}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/127931a1f73db310d9661793631ab0ec58400dc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.131ex; height:2.843ex;" alt="{\textstyle \sum a_{i}=1}"></span></td> <td><a href="/wiki/Convex_set" title="Convex set">Convex set</a></td> <td><a href="/wiki/Simplex" title="Simplex">Simplex</a> </td></tr></tbody></table> <p>Because these are more <i>restricted</i> operations, more subsets will be closed under them, so affine subsets, convex cones, and convex sets are <i>generalizations</i> of vector subspaces: a vector subspace is also an affine subspace, a convex cone, and a convex set, but a convex set need not be a vector subspace, affine, or a convex cone. </p><p>These concepts often arise when one can take certain linear combinations of objects, but not any: for example, <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a> are closed under convex combination (they form a convex set), but not conical or affine combinations (or linear), and <a href="/wiki/Positive_measure" class="mw-redirect" title="Positive measure">positive measures</a> are closed under conical combination but not affine or linear – hence one defines <a href="/wiki/Signed_measure" title="Signed measure">signed measures</a> as the linear closure. </p><p>Linear and affine combinations can be defined over any field (or ring), but conical and convex combination require a notion of "positive", and hence can only be defined over an <a href="/wiki/Ordered_field" title="Ordered field">ordered field</a> (or <a href="/wiki/Ordered_ring" title="Ordered ring">ordered ring</a>), generally the real numbers. </p><p>If one allows only scalar multiplication, not addition, one obtains a (not necessarily convex) <a href="/wiki/Cone_(linear_algebra)" class="mw-redirect" title="Cone (linear algebra)">cone</a>; one often restricts the definition to only allowing multiplication by positive scalars. </p><p>All of these concepts are usually defined as subsets of an ambient vector space (except for affine spaces, which are also considered as "vector spaces forgetting the origin"), rather than being axiomatized independently. </p> <div class="mw-heading mw-heading2"><h2 id="Operad_theory">Operad theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_combination&action=edit&section=9" title="Edit section: Operad theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Operad_theory" class="mw-redirect" title="Operad theory">Operad theory</a></div> <p>More abstractly, in the language of <a href="/wiki/Operad_theory" class="mw-redirect" title="Operad theory">operad theory</a>, one can consider vector spaces to be <a href="/wiki/Algebra_(ring_theory)" class="mw-redirect" title="Algebra (ring theory)">algebras</a> over the operad <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/673301487749dd943aa0f7cf88f146b74ab03d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.879ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{\infty }}"></span> (the infinite <a href="/wiki/Direct_sum_of_modules" title="Direct sum of modules">direct sum</a>, so only finitely many terms are non-zero; this corresponds to only taking finite sums), which parametrizes linear combinations: the vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2,3,-5,0,\dots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2,3,-5,0,\dots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee687afd19f25b304d7a94ebe045946c85bfbc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.126ex; height:2.843ex;" alt="{\displaystyle (2,3,-5,0,\dots )}"></span> for instance corresponds to the linear combination <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\mathbf {v} _{1}+3\mathbf {v} _{2}-5\mathbf {v} _{3}+0\mathbf {v} _{4}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>5</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mn>0</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\mathbf {v} _{1}+3\mathbf {v} _{2}-5\mathbf {v} _{3}+0\mathbf {v} _{4}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a74cca7eadd08f4ecc246dd4389d729e658fc4e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.595ex; height:2.509ex;" alt="{\displaystyle 2\mathbf {v} _{1}+3\mathbf {v} _{2}-5\mathbf {v} _{3}+0\mathbf {v} _{4}+\cdots }"></span>. Similarly, one can consider affine combinations, conical combinations, and convex combinations to correspond to the sub-operads where the terms sum to 1, the terms are all non-negative, or both, respectively. Graphically, these are the infinite affine hyperplane, the infinite hyper-octant, and the infinite simplex. This formalizes what is meant by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed657d7f0d7aa156e7b9b171f22b4a3aa6482c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.222ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{n}}"></span> being or the standard simplex being model spaces, and such observations as that every bounded <a href="/wiki/Convex_polytope" title="Convex polytope">convex polytope</a> is the image of a simplex. Here suboperads correspond to more restricted operations and thus more general theories. </p><p>From this point of view, we can think of linear combinations as the most general sort of operation on a vector space – saying that a vector space is an algebra over the operad of linear combinations is precisely the statement that <i>all possible</i> algebraic operations in a vector space are linear combinations. </p><p>The basic operations of addition and scalar multiplication, together with the existence of an additive identity and additive inverses, cannot be combined in any more complicated way than the generic linear combination: the basic operations are a <a href="/wiki/Generating_set" class="mw-redirect" title="Generating set">generating set</a> for the operad of all linear combinations. </p><p>Ultimately, this fact lies at the heart of the usefulness of linear combinations in the study of vector spaces. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_combination&action=edit&section=10" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>V</i> is a <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector space</a>, then there may be a way to make sense of certain <i>infinite</i> linear combinations, using the topology of <i>V</i>. For example, we might be able to speak of <i>a</i><sub>1</sub><b>v</b><sub>1</sub> + <i>a</i><sub>2</sub><b>v</b><sub>2</sub> + <i>a</i><sub>3</sub><b>v</b><sub>3</sub> + ⋯, going on forever. Such infinite linear combinations do not always make sense; we call them <i>convergent</i> when they do. Allowing more linear combinations in this case can also lead to a different concept of span, linear independence, and basis. The articles on the various flavors of topological vector spaces go into more detail about these. </p><p>If <i>K</i> is a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a> instead of a field, then everything that has been said above about linear combinations generalizes to this case without change. The only difference is that we call spaces like this <i>V</i> <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a> instead of vector spaces. If <i>K</i> is a noncommutative ring, then the concept still generalizes, with one caveat: since modules over noncommutative rings come in left and right versions, our linear combinations may also come in either of these versions, whatever is appropriate for the given module. This is simply a matter of doing scalar multiplication on the correct side. </p><p>A more complicated twist comes when <i>V</i> is a <a href="/wiki/Bimodule" title="Bimodule">bimodule</a> over two rings, <i>K</i><sub>L</sub> and <i>K</i><sub>R</sub>. In that case, the most general linear combination looks like </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\mathbf {v} _{1}b_{1}+\cdots +a_{n}\mathbf {v} _{n}b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}\mathbf {v} _{1}b_{1}+\cdots +a_{n}\mathbf {v} _{n}b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc1e6bb80267c13aac5eafdeb69106cefcaac2f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.499ex; height:2.509ex;" alt="{\displaystyle a_{1}\mathbf {v} _{1}b_{1}+\cdots +a_{n}\mathbf {v} _{n}b_{n}}"></span></dd></dl> <p>where <i>a</i><sub>1</sub>,...,<i>a</i><sub><i>n</i></sub> belong to <i>K</i><sub>L</sub>, <i>b</i><sub>1</sub>,...,<i>b</i><sub><i>n</i></sub> belong to <i>K</i><sub>R</sub>, and <b>v</b><sub>1</sub>,…,<b>v</b><sub><i>n</i></sub> belong to <i>V</i>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_combination&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Weighted_sum" class="mw-redirect" title="Weighted sum">Weighted sum</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_combination&action=edit&section=12" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFStrang2016">Strang (2016)</a> p. 3, § 1.1</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFLayLayMcDonald2016">Lay, Lay & McDonald (2016)</a> p. 28, ch. 1</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFAxler2015">Axler (2015)</a> p. 28, § 2.3</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFnLab2015">nLab (2015)</a> Linear combinations.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFAxler2015">Axler (2015)</a> pp. 29-30, §§ 2.5, 2.8</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFKatznelsonKatznelson2008">Katznelson & Katznelson (2008)</a> p. 9, § 1.2.3</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFAxler2015">Axler (2015)</a> pp. 32-33, §§ 2.17, 2.19</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFKatznelsonKatznelson2008">Katznelson & Katznelson (2008)</a> p. 14, § 1.3.2</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_combination&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Textbook">Textbook</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_combination&action=edit&section=14" title="Edit section: Textbook"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAxler2015" class="citation book cs1"><a href="/wiki/Sheldon_Axler" title="Sheldon Axler">Axler, Sheldon Jay</a> (2015). <i>Linear Algebra Done Right</i> (3rd ed.). <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media"> Springer</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-11080-6">10.1007/978-3-319-11080-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-11079-0" title="Special:BookSources/978-3-319-11079-0"><bdi>978-3-319-11079-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+Done+Right&rft.edition=3rd&rft.pub=Springer&rft.date=2015&rft_id=info%3Adoi%2F10.1007%2F978-3-319-11080-6&rft.isbn=978-3-319-11079-0&rft.aulast=Axler&rft.aufirst=Sheldon+Jay&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+combination" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatznelsonKatznelson2008" class="citation book cs1"><a href="/wiki/Yitzhak_Katznelson" title="Yitzhak Katznelson">Katznelson, Yitzhak</a>; Katznelson, Yonatan R. (2008). <i>A (Terse) Introduction to Linear Algebra</i>. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4419-9" title="Special:BookSources/978-0-8218-4419-9"><bdi>978-0-8218-4419-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+%28Terse%29+Introduction+to+Linear+Algebra&rft.pub=American+Mathematical+Society&rft.date=2008&rft.isbn=978-0-8218-4419-9&rft.aulast=Katznelson&rft.aufirst=Yitzhak&rft.au=Katznelson%2C+Yonatan+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+combination" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLayLayMcDonald2016" class="citation book cs1">Lay, David C.; Lay, Steven R.; McDonald, Judi J. (2016). <i>Linear Algebra and its Applications</i> (5th ed.). Pearson. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-98238-4" title="Special:BookSources/978-0-321-98238-4"><bdi>978-0-321-98238-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+its+Applications&rft.edition=5th&rft.pub=Pearson&rft.date=2016&rft.isbn=978-0-321-98238-4&rft.aulast=Lay&rft.aufirst=David+C.&rft.au=Lay%2C+Steven+R.&rft.au=McDonald%2C+Judi+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+combination" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrang2016" class="citation book cs1"><a href="/wiki/Gilbert_Strang" title="Gilbert Strang">Strang, Gilbert</a> (2016). <i>Introduction to Linear Algebra</i> (5th ed.). Wellesley Cambridge Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-9802327-7-6" title="Special:BookSources/978-0-9802327-7-6"><bdi>978-0-9802327-7-6</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.edition=5th&rft.pub=Wellesley+Cambridge+Press&rft.date=2016&rft.isbn=978-0-9802327-7-6&rft.aulast=Strang&rft.aufirst=Gilbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+combination" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Web">Web</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_combination&action=edit&section=15" title="Edit section: Web"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFnLab2015" class="citation web cs1"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/linear+combination">"Linear Combinations"</a>. <i><a href="/wiki/NLab" title="NLab">nLab</a></i>. 27 October 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">16 Feb</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=nLab&rft.atitle=Linear+Combinations&rft.date=2015-10-27&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2Flinear%2Bcombination&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALinear+combination" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Linear_combination&action=edit&section=16" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/linear_combinations/v/linear-combinations-and-span">Linear Combinations and Span: Understanding linear combinations and spans of vectors</a>, 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href="/wiki/Outline_of_linear_algebra" title="Outline of linear algebra">Outline</a></li> <li><a href="/wiki/Glossary_of_linear_algebra" title="Glossary of linear algebra">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">Scalar</a></li> <li><a href="/wiki/Euclidean_vector" title="Euclidean vector">Vector</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li> <li><a href="/wiki/Scalar_multiplication" title="Scalar multiplication">Scalar multiplication</a></li> <li><a href="/wiki/Vector_projection" title="Vector projection">Vector projection</a></li> <li><a href="/wiki/Linear_span" title="Linear span">Linear span</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a 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title="Transpose">Transpose</a></li> <li><a href="/wiki/System_of_linear_equations" title="System of linear equations">Linear equations</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Euclidean_space" title="Euclidean space"><img alt="Three dimensional Euclidean space" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/80px-Linear_subspaces_with_shading.svg.png" decoding="async" width="80" height="58" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/120px-Linear_subspaces_with_shading.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/160px-Linear_subspaces_with_shading.svg.png 2x" data-file-width="325" data-file-height="236" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrices</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Block_matrix" title="Block matrix">Block</a></li> <li><a href="/wiki/Matrix_decomposition" title="Matrix decomposition">Decomposition</a></li> <li><a href="/wiki/Invertible_matrix" title="Invertible matrix">Invertible</a></li> <li><a href="/wiki/Minor_(linear_algebra)" title="Minor (linear algebra)">Minor</a></li> <li><a href="/wiki/Matrix_multiplication" title="Matrix multiplication">Multiplication</a></li> <li><a href="/wiki/Rank_(linear_algebra)" title="Rank (linear algebra)">Rank</a></li> <li><a href="/wiki/Transformation_matrix" title="Transformation matrix">Transformation</a></li> <li><a href="/wiki/Cramer%27s_rule" title="Cramer's rule">Cramer's rule</a></li> <li><a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a></li> <li><a href="/wiki/Productive_matrix" title="Productive matrix">Productive matrix</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Bilinear_map" title="Bilinear map">Bilinear</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Orthogonality" title="Orthogonality">Orthogonality</a></li> <li><a href="/wiki/Dot_product" title="Dot product">Dot product</a></li> <li><a href="/wiki/Hadamard_product_(matrices)" title="Hadamard product (matrices)">Hadamard product</a></li> <li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a></li> <li><a href="/wiki/Outer_product" title="Outer product">Outer product</a></li> <li><a href="/wiki/Kronecker_product" title="Kronecker product">Kronecker product</a></li> <li><a href="/wiki/Gram%E2%80%93Schmidt_process" title="Gram–Schmidt process">Gram–Schmidt process</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Determinant" title="Determinant">Determinant</a></li> <li><a href="/wiki/Cross_product" title="Cross product">Cross product</a></li> <li><a href="/wiki/Triple_product" title="Triple product">Triple product</a></li> <li><a href="/wiki/Seven-dimensional_cross_product" title="Seven-dimensional cross product">Seven-dimensional cross product</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a></li> <li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior algebra</a></li> <li><a href="/wiki/Bivector" title="Bivector">Bivector</a></li> <li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li> <li><a href="/wiki/Outermorphism" title="Outermorphism">Outermorphism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_space" title="Vector space">Vector space</a> constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dual_space" title="Dual space">Dual</a></li> <li><a href="/wiki/Direct_sum_of_modules#Construction_for_two_vector_spaces" title="Direct sum of modules">Direct sum</a></li> <li><a href="/wiki/Function_space#In_linear_algebra" title="Function space">Function space</a></li> <li><a href="/wiki/Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">Quotient</a></li> <li><a href="/wiki/Linear_subspace" title="Linear subspace">Subspace</a></li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li></ul> </div></td></tr><tr><th scope="row" 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Linear combination - Wikipedia