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[o]" accesskey="o" class=""><span>로그인</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="더 많은 옵션" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="개인 도구" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">개인 도구</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="사용자 메뉴" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ko.wikipedia.org&uselang=ko"><span>기부</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%EC%84%A0%ED%98%95+%EA%B2%B0%ED%95%A9" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>계정 만들기</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EC%84%A0%ED%98%95+%EA%B2%B0%ED%95%A9" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>로그인</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 로그아웃한 편집자를 위한 문서 <a href="/wiki/%EB%8F%84%EC%9B%80%EB%A7%90:%EC%86%8C%EA%B0%9C" aria-label="편집에 관해 더 알아보기"><span>더 알아보기</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EA%B8%B0%EC%97%AC" title="이 IP 주소의 편집 목록 [y]" accesskey="y"><span>기여</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EC%82%AC%EC%9A%A9%EC%9E%90%ED%86%A0%EB%A1%A0" title="현재 사용하는 IP 주소에 대한 토론 문서 [n]" accesskey="n"><span>토론</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="사이트"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="목차" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">목차</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">숨기기</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">처음 위치</div> </a> </li> <li id="toc-정의" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#정의"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>정의</span> </div> </a> <ul id="toc-정의-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-선형생성" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#선형생성"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>선형생성</span> </div> </a> <ul id="toc-선형생성-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-선형_독립" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#선형_독립"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>선형 독립</span> </div> </a> <ul id="toc-선형_독립-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-다양한_종류의_선형_결합" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#다양한_종류의_선형_결합"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>다양한 종류의 선형 결합</span> </div> </a> <ul id="toc-다양한_종류의_선형_결합-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-일반화" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#일반화"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>일반화</span> </div> </a> <ul id="toc-일반화-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-각주" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#각주"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>각주</span> </div> </a> <ul id="toc-각주-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-참고_문헌" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#참고_문헌"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>참고 문헌</span> </div> </a> <button aria-controls="toc-참고_문헌-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>참고 문헌 하위섹션 토글하기</span> </button> <ul id="toc-참고_문헌-sublist" class="vector-toc-list"> <li id="toc-교과서" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#교과서"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>교과서</span> </div> </a> <ul id="toc-교과서-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-웹" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#웹"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>웹</span> </div> </a> <ul id="toc-웹-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-외부_링크" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#외부_링크"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>외부 링크</span> </div> </a> <ul id="toc-외부_링크-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="목차" 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="목차 토글" > <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">목차 토글</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">선형 결합</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="다른 언어로 문서를 방문합니다. 36개 언어로 읽을 수 있습니다" > <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">36개 언어</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 – 아랍어" lang="ar" hreflang="ar" data-title="تركيب خطي" data-language-autonym="العربية" data-language-local-name="아랍어" 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 – 카탈로니아어" lang="ca" hreflang="ca" data-title="Combinació lineal" data-language-autonym="Català" data-language-local-name="카탈로니아어" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Line%C3%A1rn%C3%AD_kombinace" title="Lineární kombinace – 체코어" lang="cs" hreflang="cs" data-title="Lineární kombinace" data-language-autonym="Čeština" data-language-local-name="체코어" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B8%D0%BB%D0%BB%D0%B5_%D0%BA%D0%BE%D0%BC%D0%B1%D0%B8%D0%BD%D0%B0%D1%86%D0%B8" title="Линилле комбинаци – 추바시어" lang="cv" hreflang="cv" data-title="Линилле комбинаци" data-language-autonym="Чӑвашла" data-language-local-name="추바시어" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Linearkombination" title="Linearkombination – 독일어" lang="de" hreflang="de" data-title="Linearkombination" data-language-autonym="Deutsch" data-language-local-name="독일어" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Linear_combination" title="Linear combination – 영어" lang="en" hreflang="en" data-title="Linear combination" data-language-autonym="English" data-language-local-name="영어" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Lineara_kombina%C4%B5o" title="Lineara kombinaĵo – 에스페란토어" lang="eo" hreflang="eo" data-title="Lineara kombinaĵo" data-language-autonym="Esperanto" data-language-local-name="에스페란토어" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Combinaci%C3%B3n_lineal" title="Combinación lineal – 스페인어" lang="es" hreflang="es" data-title="Combinación lineal" data-language-autonym="Español" data-language-local-name="스페인어" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%B1%DA%A9%DB%8C%D8%A8_%D8%AE%D8%B7%DB%8C" title="ترکیب خطی – 페르시아어" lang="fa" hreflang="fa" data-title="ترکیب خطی" data-language-autonym="فارسی" data-language-local-name="페르시아어" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Lineaarikombinaatio" title="Lineaarikombinaatio – 핀란드어" lang="fi" hreflang="fi" data-title="Lineaarikombinaatio" data-language-autonym="Suomi" data-language-local-name="핀란드어" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Combinaison_lin%C3%A9aire" title="Combinaison linéaire – 프랑스어" lang="fr" hreflang="fr" data-title="Combinaison linéaire" data-language-autonym="Français" data-language-local-name="프랑스어" 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 – 갈리시아어" lang="gl" hreflang="gl" data-title="Combinación linear" data-language-autonym="Galego" data-language-local-name="갈리시아어" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%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 – 히브리어" lang="he" hreflang="he" data-title="צירוף ליניארי" data-language-autonym="עברית" data-language-local-name="히브리어" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Linearna_kombinacija" title="Linearna kombinacija – 크로아티아어" lang="hr" hreflang="hr" data-title="Linearna kombinacija" data-language-autonym="Hrvatski" data-language-local-name="크로아티아어" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Line%C3%A1ris_kombin%C3%A1ci%C3%B3" title="Lineáris kombináció – 헝가리어" lang="hu" hreflang="hu" data-title="Lineáris kombináció" data-language-autonym="Magyar" data-language-local-name="헝가리어" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kombinasi_linear" title="Kombinasi linear – 인도네시아어" lang="id" hreflang="id" data-title="Kombinasi linear" data-language-autonym="Bahasa Indonesia" data-language-local-name="인도네시아어" 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 – 이탈리아어" lang="it" hreflang="it" data-title="Combinazione lineare" data-language-autonym="Italiano" data-language-local-name="이탈리아어" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B7%9A%E5%9E%8B%E7%B5%90%E5%90%88" title="線型結合 – 일본어" lang="ja" hreflang="ja" data-title="線型結合" data-language-autonym="日本語" data-language-local-name="일본어" class="interlanguage-link-target"><span>日本語</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 – 말라얄람어" lang="ml" hreflang="ml" data-title="രേഖീയസഞ്ചയം" data-language-autonym="മലയാളം" data-language-local-name="말라얄람어" 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 – 네덜란드어" lang="nl" hreflang="nl" data-title="Lineaire combinatie" data-language-autonym="Nederlands" data-language-local-name="네덜란드어" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Line%C3%A6rkombinasjon" title="Lineærkombinasjon – 노르웨이어(보크말)" lang="nb" hreflang="nb" data-title="Lineærkombinasjon" data-language-autonym="Norsk bokmål" data-language-local-name="노르웨이어(보크말)" 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 – 폴란드어" lang="pl" hreflang="pl" data-title="Kombinacja liniowa" data-language-autonym="Polski" data-language-local-name="폴란드어" 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 – 포르투갈어" lang="pt" hreflang="pt" data-title="Combinação linear" data-language-autonym="Português" data-language-local-name="포르투갈어" 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 – 러시아어" lang="ru" hreflang="ru" data-title="Линейная комбинация" data-language-autonym="Русский" data-language-local-name="러시아어" 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 – 슬로바키아어" lang="sk" hreflang="sk" data-title="Lineárna kombinácia" data-language-autonym="Slovenčina" data-language-local-name="슬로바키아어" 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 – 슬로베니아어" lang="sl" hreflang="sl" data-title="Linearna kombinacija" data-language-autonym="Slovenščina" data-language-local-name="슬로베니아어" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Linj%C3%A4rkombination" title="Linjärkombination – 스웨덴어" lang="sv" hreflang="sv" data-title="Linjärkombination" data-language-autonym="Svenska" data-language-local-name="스웨덴어" 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 – 타밀어" lang="ta" hreflang="ta" data-title="நேரியல் சேர்வு" data-language-autonym="தமிழ்" data-language-local-name="타밀어" 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 – 태국어" lang="th" hreflang="th" data-title="ผลรวมเชิงเส้น" data-language-autonym="ไทย" data-language-local-name="태국어" 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 – 터키어" lang="tr" hreflang="tr" data-title="Doğrusal birleşim" 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가리키는 링크 [g]" accesskey="g"><span>위키데이터 항목</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="페이지 도구"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="보이기"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">보이기</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">숨기기</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">위키백과, 우리 모두의 백과사전.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ko" dir="ltr"><p><span class="nowrap"></span> <b>선형 결합</b>(線型結合, <span lang="en">linear combination</span>) 또는 <b>일차 결합</b>(一次結合)은 <a href="/wiki/%EC%88%98%ED%95%99" title="수학">수학</a>에서 각 항에 <a href="/wiki/%EC%83%81%EC%88%98" title="상수">상수</a>를 곱하고 결과를 더함으로써 일련의 항으로 구성된 표현식이다(예: <i>x</i>와 <i>y</i>의 선형 결합은 <i>ax</i> + <i>by</i> 형식인데 여기서 <i>a</i>와 <i>b</i>는 상수이다).<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> 선형 결합의 개념은 <a href="/wiki/%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99" title="선형대수학">선형대수학</a>과 수학 관련 분야의 중심이다. 이 글의 대부분은 <a href="/wiki/%EC%B2%B4_(%EC%88%98%ED%95%99)" title="체 (수학)">체</a> 위의 벡터 공간의 맥락에서 선형 결합을 다루며 글의 끝에 주어진 일부 일반화를 다룬다. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="정의"><span id=".EC.A0.95.EC.9D.98"></span>정의</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%84%A0%ED%98%95_%EA%B2%B0%ED%95%A9&action=edit&section=1" title="부분 편집: 정의"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>V</i>를 체 <i>K</i> 위의 <a href="/wiki/%EB%B2%A1%ED%84%B0_%EA%B3%B5%EA%B0%84" title="벡터 공간">벡터 공간</a>이 되도록 한다. 우리는 평소와 같이 <i>V</i> 벡터 공간의 원소를 부르고 <i>K</i> <a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC_(%EC%88%98%ED%95%99)" title="스칼라 (수학)">스칼라</a>의 원소를 부른다. 만약 <b>v</b><sub>1</sub>,...,<b>v</b><sub><i>n</i></sub>이 벡터이고 <i>a</i><sub>1</sub>,...,<i>a</i><sub><i>n</i></sub>이 스칼라인 경우에는 해당 스칼라와 계수의 선형 결합은 <span class="mwe-math-element"><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> </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/9c8a02a1e523b25d0e9a8c2fc5d0da3b096c44d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.41ex; 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>이라고 표현한다. </p><p>"선형 결합"이라는 용어가 표현식을 참조하는지 또는 그 값을 참조하는지 여부에 대해서는 다소 애매한 점이 있다. 대부분의 경우 "<b>v</b><sub>1</sub>,...,<b>v</b><sub><i>n</i></sub>은 항상 하위 공간을 형성한다"는 주장에서처럼 값이 강조된다. 그러나 "두 개의 서로 다른 선형 결합이 동일한 값을 가질 수 있다"고 말할 수 있으며 이 경우에는 표현식에 대한 참조가 된다. 이러한 용도들 사이의 미묘한 차이는 선형 종속 집합의 본질이다. 벡터족 <i>F</i>는 값으로서 <i>F</i>의 벡터의 선형 결합이 고유하게 식과 같은 경우 정확하게 선형 독립적이다. 어떤 경우든 표현식으로 볼 때조차 선형 결합에 대해 중요한 것은 각 <b>v</b><sub><i>i</i></sub>의 계수 뿐이다. 항을 허용하거나 영점 계수를 갖는 항을 추가하는 것과 같은 사소한 수정은 뚜렷한 선형 결합을 생성하지 않는다. </p><p>주어진 상황에서 <i>K</i>와 <i>V</i>는 명시적으로 지정되거나 문맥상 명백할 수 있다. 이 경우 우리는 종종 계수 <b>v</b><sub>1</sub>,...,<b>v</b><sub><i>n</i></sub>의 선형 결합(반드시 <i>K</i>에 속해야 한다는 점 제외)을 언급한다. 또는 <i>S</i>가 <i>V</i>의 <a href="/wiki/%EB%B6%80%EB%B6%84%EC%A7%91%ED%95%A9" title="부분집합">부분집합</a>인 경우 벡터가 집합 <i>S</i>에 속해야 한다는 점을 제외하고 계수와 벡터가 모두 지정되지 않은 <i>S</i>에서 벡터의 선형 결합에 대해 말할 수 있다. 마지막으로 우리는 벡터가 <i>V</i>에 속해야 하고 계수가 <i>K</i>에 속해야 한다는 것을 제외하면 아무것도 지정되지 않은 선형 결합에 대해 간단히 말할 수 있다. 이 경우 <i>V</i>의 모든 벡터는 확실히 어떤 선형 결합의 값이기 때문에 한 가지는 아마도 식을 참조할 것이다. </p><p>선형 결합은 정의상 매우 많은 벡터만 포함한다(아래와 같이 언급된 <b>일반화</b>인 경우는 제외). 그러나 벡터가 하나를 언급할 경우에는 추출된 집합 <i>S</i>는 여전히 무한할 수 있다. 각각의 개별 선형 결합은 단지 많은 벡터를 포함할 뿐이다. 또한 <i>n</i>이 0이 될 수 없는 이유는 없다. 이 경우 우리는 관례에 따라 선형 결합의 결과가 <i>V</i>의 제로 벡터(zero vector)라고 선언한다. </p> <div class="mw-heading mw-heading2"><h2 id="선형생성"><span id=".EC.84.A0.ED.98.95.EC.83.9D.EC.84.B1"></span>선형생성</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%84%A0%ED%98%95_%EA%B2%B0%ED%95%A9&action=edit&section=2" title="부분 편집: 선형생성"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r34311305">.mw-parser-output .hatnote{}.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}</style><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EC%84%A0%ED%98%95%EC%83%9D%EC%84%B1" class="mw-redirect" title="선형생성">선형생성</a>입니다.</div> <p>임의의 체 <i>K</i>, 임의의 벡터 공간 <i>V</i>를 사용하고 <b>v</b><sub>1</sub>,...,<b>v</b><sub><i>n</i></sub>을 벡터(<i>V</i> 단위)로 설정합니다. 이러한 벡터의 모든 선형 결합 집합을 고려하는 것은 흥미로운 편이다. 이 집합을 벡터의 <a href="/wiki/%EC%84%A0%ED%98%95%EC%83%9D%EC%84%B1" class="mw-redirect" title="선형생성">선형생성</a>(예: <i>S</i> = {<b>v</b><sub>1</sub>, ..., <b>v</b><sub><i>n</i></sub>})이라고 부른다. 우리는 <i>S</i>의 생성 범위인 span(<i>S</i>) 또는 sp(<i>S</i>)를 다음과 같이 표현한다.<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> </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> </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/e2632189f6b10a361979e54b7c48c99a1383d61b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.801ex; 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="선형_독립"><span id=".EC.84.A0.ED.98.95_.EB.8F.85.EB.A6.BD"></span>선형 독립</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%84%A0%ED%98%95_%EA%B2%B0%ED%95%A9&action=edit&section=3" title="부분 편집: 선형 독립"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EC%9D%BC%EC%B0%A8_%EB%8F%85%EB%A6%BD_%EC%A7%91%ED%95%A9" title="일차 독립 집합">일차 독립 집합</a>입니다.</div> <p>일부 벡터 <b>v</b><sub>1</sub>,...,<b>v</b><sub><i>n</i></sub>의 경우 단일 벡터는 2 가지 다른 방법으로 선형 결합으로 쓸 수 있다. </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>이와 마찬가지로 (<span class="mwe-math-element"><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>)를 빼면 중요하지 않은 결합은 영점이 된다.<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>이 경우 <b>v</b><sub>1</sub>,...,<b>v</b><sub><i>n</i></sub>는 <b>선형 종속</b>이 되지만 그렇지 않으면 <b>선형 독립</b>(<a href="/wiki/%EC%9D%BC%EC%B0%A8_%EB%8F%85%EB%A6%BD_%EC%A7%91%ED%95%A9" title="일차 독립 집합">일차 독립 집합</a>)이라고 부른다. 마찬가지로 우리는 벡터의 임의 집합 <i>S</i>의 선형 종속 또는 독립에 대해 말할 수 있다. </p><p><i>S</i>가 선형 독립적이고 <i>S</i>의 범위가 <i>V</i>와 같으면 <i>S</i>는 <i>V</i>의 <a href="/wiki/%EA%B8%B0%EC%A0%80_(%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99)" title="기저 (선형대수학)">기저</a>가 된다. </p> <div class="mw-heading mw-heading2"><h2 id="다양한_종류의_선형_결합"><span id=".EB.8B.A4.EC.96.91.ED.95.9C_.EC.A2.85.EB.A5.98.EC.9D.98_.EC.84.A0.ED.98.95_.EA.B2.B0.ED.95.A9"></span>다양한 종류의 선형 결합</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%84%A0%ED%98%95_%EA%B2%B0%ED%95%A9&action=edit&section=4" title="부분 편집: 다양한 종류의 선형 결합"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>선형 결합에 사용되는 계수를 제한함으로써 아핀 결합, 원뿔 결합 및 볼록 결합의 관련 개념과 이러한 연산에 따라 닫힌 집합의 관련 개념을 정의할 수 있다. </p> <table class="wikitable" border="1" style="text-align: left;"> <tbody><tr> <th>결합 유형</th> <th>계수 제한</th> <th>집합 이름</th> <th>공간의 모델 </th></tr> <tr> <td>선형 결합</td> <td>제한 없음</td> <td>벡터 부분 공간</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>아핀 결합</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>아핀 부분 공간</td> <td>아핀 <a href="/wiki/%EC%B4%88%ED%8F%89%EB%A9%B4_(%EC%88%98%ED%95%99)" title="초평면 (수학)">초평면</a> </td></tr> <tr> <td>원뿔 결합</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>볼록 원뿔</td> <td><a href="/wiki/%EC%82%AC%EB%B6%84%EB%A9%B4" title="사분면">사분면</a>, 팔분원, 사분면 </td></tr> <tr> <td><a href="/w/index.php?title=%EB%B3%BC%EB%A1%9D_%EA%B2%B0%ED%95%A9&action=edit&redlink=1" class="new" title="볼록 결합 (없는 문서)">볼록 결합</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/%EB%B3%BC%EB%A1%9D_%EC%A7%91%ED%95%A9" title="볼록 집합">볼록 집합</a></td> <td><a href="/wiki/%EB%8B%A8%EC%B2%B4_(%EC%88%98%ED%95%99)" title="단체 (수학)">단체</a> </td></tr></tbody></table> <p>이들은 보다 제한된 연산이기 때문에 더 많은 부분집합들이 그들 아래쪽에서 닫힐 것이기 때문에 아핀 부분집합, 볼록 원뿔, 볼록 집합은 벡터 하위 공간의 일반화이다. 또한 벡터 부분공간은 아핀 부분 공간, 볼록 원뿔, 볼록 집합이지만 볼록 집합은 벡터 하위 공간, 아핀 또는 볼록 원뿔일 필요가 없다. </p><p>이러한 개념은 물체의 특정한 선형 결합을 취할 수 있을 때에 종종 발생하지만 어떤 것도 발생하지 않는다. 예를 들어 확률 분포는 볼록 결합(볼록 집합을 형성함)에서는 닫히지만 원뿔 또는 아핀 결합(또는 선형)에서는 닫히지 않으며 양의 측정은 원뿔 결합에서는 닫히지만 아핀 또는 선형은 되지 않는다. 따라서 하나의 정의라고 표현할 수 있는데 선형 폐쇄로 서명된 측정값을 나타냅니다. </p><p>선형 및 아핀 결합은 모든 체(또는 환)에 대해 정의될 수 있지만 원뿔 및 볼록 결합은 "양수"의 개념을 필요로 하므로 정렬된 체(또는 정렬된 환), 일반적으로 실수에 대해서만 정의될 수 있다. 덧셈이 아니라 스칼라 곱셈만 허용하면(꼭 볼록하지는 않은) 원뿔을 얻을 수 있다. 종종 양수인 스칼라에 의한 곱셈만 허용하도록 정의를 제한한다. 이러한 모든 개념은 일반적으로 독립적으로 공리화되기 보다는 주변 벡터 공간의 부분 집합(아핀 공간, "원점을 잊은 벡터 공간"이라고도 간주되는 공간 제외)으로 정의된다. </p> <div class="mw-heading mw-heading2"><h2 id="일반화"><span id=".EC.9D.BC.EB.B0.98.ED.99.94"></span>일반화</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%84%A0%ED%98%95_%EA%B2%B0%ED%95%A9&action=edit&section=5" title="부분 편집: 일반화"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>V</i>가 <a href="/wiki/%EC%9C%84%EC%83%81_%EB%B2%A1%ED%84%B0_%EA%B3%B5%EA%B0%84" title="위상 벡터 공간">위상 벡터 공간</a>이라면 <i>V</i>의 <a href="/wiki/%EC%9C%84%EC%83%81%EC%88%98%ED%95%99" title="위상수학">위상수학</a>을 사용하여 특정 무한 선형 결합을 이해할 수 있는 방법이 있을 수 있다. 예를 들어 우리는 <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> + ⋯가 영원하다고 말할 수 있다. 그러한 무한 선형 결합은 항상 이치에 맞는 것은 아니다. 우리는 이 결합을 <b>융합</b>이라고 부른다. 이 경우 더 많은 선형 결합을 허용하면 생성, 선형 독립 및 기저의 다른 개념으로 이어질 수도 있다. </p><p>만약 <i>K</i>가 체 대신에 <a href="/wiki/%EA%B0%80%ED%99%98%ED%99%98" title="가환환">가환환</a>이라면 선형 결합에 대해 위에서 말한 모든 것은 변하지 않고 이 경우에 일반화된다. 유일한 차이점은 벡터 공간 대신에 이러한 <i>V</i> <a href="/wiki/%EA%B0%80%EA%B5%B0" title="가군">가군</a>과 같은 공간을 부른다는 것이다. K가 비가환환인 경우에도 개념은 여전히 일반화되며 한 가지 주의 사항은 다음과 같다. 비가환환 위에 있는 모듈들은 왼쪽과 오른쪽 버전으로 나오기 때문에 선형 결합은 주어진 모듈에 적절한 어떤 것이든 이러한 버전들 가운데 하나에서도 나올 수 있다. 이는 단순히 스칼라 곱셈을 올바른 측면에서 수행하는 문제이다. </p><p>더 복잡한 반전은 <i>V</i>가 <i>K</i><sub>L</sub>과 <i>K</i><sub>R</sub> 2개의 환 위에 있는 <a href="/wiki/%EC%8C%8D%EA%B0%80%EA%B5%B0" title="쌍가군">쌍가군</a>일 때에 발생한다. 이 경우에 가장 일반적인 선형 결합은 다음과 같다. </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>여기서 <i>a</i><sub>1</sub>,...,<i>a</i><sub><i>n</i></sub>은 <i>K</i><sub>L</sub>, <i>b</i><sub>1</sub>,...,<i>b</i><sub><i>n</i></sub>은 <i>K</i><sub>R</sub>, <b>v</b><sub>1</sub>,…,<b>v</b><sub><i>n</i></sub>은 <i>V</i>에 속한다. </p> <div class="mw-heading mw-heading2"><h2 id="각주"><span id=".EA.B0.81.EC.A3.BC"></span>각주</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%84%A0%ED%98%95_%EA%B2%B0%ED%95%A9&action=edit&section=6" title="부분 편집: 각주"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r35556958">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.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"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">(<a href="#CITEREFStrang2016">Strang 2016</a>)<span class="error harv-error" style="display: none; font-size:100%"> harv error: 대상 없음: CITEREFStrang2016 (<a href="https://en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" class="extiw" title="en:Category:Harv and Sfn template errors">help</a>)</span> p. 3, § 1.1</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">(<a href="#CITEREFLayLayMcDonald2016">Lay, Lay & McDonald 2016</a>)<span class="error harv-error" style="display: none; font-size:100%"> harv error: 대상 없음: CITEREFLayLayMcDonald2016 (<a href="https://en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" class="extiw" title="en:Category:Harv and Sfn template errors">help</a>)</span> p. 28, ch. 1</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">(<a href="#CITEREFAxler2015">Axler 2015</a>)<span class="error harv-error" style="display: none; font-size:100%"> harv error: 대상 없음: CITEREFAxler2015 (<a href="https://en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" class="extiw" title="en:Category:Harv and Sfn template errors">help</a>)</span> p. 28, § 2.3</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></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"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">(<a href="#CITEREFAxler2015">Axler 2015</a>)<span class="error harv-error" style="display: none; font-size:100%"> harv error: 대상 없음: CITEREFAxler2015 (<a href="https://en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" class="extiw" title="en:Category:Harv and Sfn template errors">help</a>)</span> pp. 29-30, §§ 2.5, 2.8</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">(<a href="#CITEREFKatznelsonKatznelson2008">Katznelson & Katznelson 2008</a>)<span class="error harv-error" style="display: none; font-size:100%"> harv error: 대상 없음: CITEREFKatznelsonKatznelson2008 (<a href="https://en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" class="extiw" title="en:Category:Harv and Sfn template errors">help</a>)</span> p. 9, § 1.2.3</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text">(<a href="#CITEREFAxler2015">Axler 2015</a>)<span class="error harv-error" style="display: none; font-size:100%"> harv error: 대상 없음: CITEREFAxler2015 (<a href="https://en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" class="extiw" title="en:Category:Harv and Sfn template errors">help</a>)</span> pp. 32-33, §§ 2.17, 2.19</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text">(<a href="#CITEREFKatznelsonKatznelson2008">Katznelson & Katznelson 2008</a>)<span class="error harv-error" style="display: none; font-size:100%"> harv error: 대상 없음: CITEREFKatznelsonKatznelson2008 (<a href="https://en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" class="extiw" title="en:Category:Harv and Sfn template errors">help</a>)</span> p. 14, § 1.3.2</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="참고_문헌"><span id=".EC.B0.B8.EA.B3.A0_.EB.AC.B8.ED.97.8C"></span>참고 문헌</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%84%A0%ED%98%95_%EA%B2%B0%ED%95%A9&action=edit&section=7" title="부분 편집: 참고 문헌"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="교과서"><span id=".EA.B5.90.EA.B3.BC.EC.84.9C"></span>교과서</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%84%A0%ED%98%95_%EA%B2%B0%ED%95%A9&action=edit&section=8" title="부분 편집: 교과서"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation book">Axler, Sheldon Jay (2015년). 《Linear Algebra Done Right》 3판. Springer. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-3-319-11079-0" title="특수:책찾기/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=3&rft.pub=Springer&rft.date=2015&rft.isbn=978-3-319-11079-0&rft.aulast=Axler&rft.aufirst=Sheldon+Jay&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%84%A0%ED%98%95+%EA%B2%B0%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Katznelson, Yitzhak; Katznelson, Yonatan R. (2008년). 《A (Terse) Introduction to Linear Algebra》. American Mathematical Society. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-0-8218-4419-9" title="특수:책찾기/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%2Fko.wikipedia.org%3A%EC%84%A0%ED%98%95+%EA%B2%B0%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Lay, David C.; Lay, Steven R.; McDonald, Judi J. (2016년). 《Linear Algebra and its Applications》 5판. Pearson. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-0-321-98238-4" title="특수:책찾기/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=5&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%2Fko.wikipedia.org%3A%EC%84%A0%ED%98%95+%EA%B2%B0%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Strang, Gilbert (2016년). 《Introduction to Linear Algebra》 5판. Wellesley Cambridge Press. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-0-9802327-7-6" title="특수:책찾기/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=5&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%2Fko.wikipedia.org%3A%EC%84%A0%ED%98%95+%EA%B2%B0%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="웹"><span id=".EC.9B.B9"></span>웹</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%84%A0%ED%98%95_%EA%B2%B0%ED%95%A9&action=edit&section=9" title="부분 편집: 웹"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFnLab2015" class="citation web"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/linear+combination">“Linear Combinations”</a>. 《nLab》. 2015년 10월 27일<span class="reference-accessdate">. 2021년 2월 16일에 확인함</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%2Fko.wikipedia.org%3A%EC%84%A0%ED%98%95+%EA%B2%B0%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="외부_링크"><span id=".EC.99.B8.EB.B6.80_.EB.A7.81.ED.81.AC"></span>외부 링크</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%84%A0%ED%98%95_%EA%B2%B0%ED%95%A9&action=edit&section=10" title="부분 편집: 외부 링크"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/linear_combinations/v/linear-combinations-and-span">Linear Combinations and Span: Understanding linear combinations and spans of vectors</a>, khanacademy.org.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r36480591">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl 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href="/w/index.php?title=%ED%8B%80%ED%86%A0%EB%A1%A0:%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99&action=edit&redlink=1" class="new" title="틀토론:선형대수학 (없는 문서)"><abbr title="이 틀에 관해 토론하기" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">t</abbr></a></li><li class="nv-편집"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%AC%B8%EC%84%9C%ED%8E%B8%EC%A7%91/%ED%8B%80:%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99" title="특수:문서편집/틀:선형대수학"><abbr title="이 틀을 편집하기" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">e</abbr></a></li></ul></div><div id="선형대수학" style="font-size:114%;margin:0 4em"><a href="/wiki/%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99" title="선형대수학">선형대수학</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">기본 개념</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC_(%EC%88%98%ED%95%99)" title="스칼라 (수학)">스칼라</a></li> <li><a href="/wiki/%EB%B2%A1%ED%84%B0_(%EB%AC%BC%EB%A6%AC)" class="mw-redirect" title="벡터 (물리)">벡터</a></li> <li><a href="/wiki/%EB%B2%A1%ED%84%B0_%EA%B3%B5%EA%B0%84" title="벡터 공간">벡터 공간</a></li> <li><a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC_%EA%B3%B1%EC%85%88" title="스칼라 곱셈">스칼라 곱셈</a></li> <li><a href="/w/index.php?title=%EB%B2%A1%ED%84%B0_%EC%82%AC%EC%98%81&action=edit&redlink=1" class="new" title="벡터 사영 (없는 문서)">벡터 사영</a></li> <li><a href="/wiki/%EC%84%A0%ED%98%95%EC%83%9D%EC%84%B1" class="mw-redirect" title="선형생성">선형생성</a></li> <li><a href="/wiki/%EC%84%A0%ED%98%95_%EB%B3%80%ED%99%98" title="선형 변환">선형 변환</a></li> <li><a href="/wiki/%EC%82%AC%EC%98%81%EC%9E%91%EC%9A%A9%EC%86%8C" title="사영작용소">사영작용소</a></li> <li><a href="/wiki/%EC%9D%BC%EC%B0%A8_%EB%8F%85%EB%A6%BD_%EC%A7%91%ED%95%A9" title="일차 독립 집합">일차 독립 집합</a></li> <li><a class="mw-selflink selflink">선형 결합</a></li> <li><a href="/wiki/%EA%B8%B0%EC%A0%80_(%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99)" title="기저 (선형대수학)">기저</a></li> <li><a href="/w/index.php?title=%EA%B8%B0%EC%A0%80_%EB%B3%80%EA%B2%BD&action=edit&redlink=1" class="new" title="기저 변경 (없는 문서)">기저 변경</a></li> <li><a href="/w/index.php?title=%ED%96%89_%EB%B2%A1%ED%84%B0%EC%99%80_%EC%97%B4_%EB%B2%A1%ED%84%B0&action=edit&redlink=1" class="new" title="행 벡터와 열 벡터 (없는 문서)">행 벡터와 열 벡터</a></li> <li><a href="/w/index.php?title=%ED%96%89_%EA%B3%B5%EA%B0%84%EA%B3%BC_%EC%97%B4_%EA%B3%B5%EA%B0%84&action=edit&redlink=1" class="new" title="행 공간과 열 공간 (없는 문서)">행 공간과 열 공간</a></li> <li><a href="/wiki/%EC%A7%81%EA%B5%90" title="직교">직교</a></li> <li><a href="/wiki/%EC%98%81%EA%B3%B5%EA%B0%84" class="mw-redirect" title="영공간">영공간</a></li> <li><a href="/wiki/%EA%B3%A0%EC%9C%B3%EA%B0%92%EA%B3%BC_%EA%B3%A0%EC%9C%A0_%EB%B2%A1%ED%84%B0" title="고윳값과 고유 벡터">고윳값과 고유 벡터</a></li> <li><a href="/wiki/%EC%99%B8%EC%A0%81" title="외적">외적</a></li> <li><a href="/wiki/%EB%82%B4%EC%A0%81_%EA%B3%B5%EA%B0%84" title="내적 공간">내적 공간</a></li> <li><a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1" title="스칼라곱">스칼라곱</a></li> <li><a href="/wiki/%EC%A0%84%EC%B9%98%ED%96%89%EB%A0%AC" class="mw-redirect" title="전치행렬">전치행렬</a></li> <li><a href="/wiki/%EA%B7%B8%EB%9E%8C-%EC%8A%88%EB%AF%B8%ED%8A%B8_%EA%B3%BC%EC%A0%95" title="그람-슈미트 과정">그람-슈미트 과정</a></li> <li><a href="/wiki/%EC%97%B0%EB%A6%BD_%EC%9D%BC%EC%B0%A8_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="연립 일차 방정식">일차 방정식</a></li> <li><a href="/w/index.php?title=%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99%EC%9D%98_%EA%B8%B0%EB%B3%B8_%EC%A0%95%EB%A6%AC&action=edit&redlink=1" class="new" title="선형대수학의 기본 정리 (없는 문서)">기본 정리</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" 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href="/wiki/%EB%8B%A4%EC%A4%91%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99" title="다중선형대수학">다중선형대수학</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EA%B8%B0%ED%95%98%EC%A0%81_%EB%8C%80%EC%88%98%ED%95%99" title="기하적 대수학">기하적 대수학</a></li> <li><a href="/wiki/%EC%99%B8%EB%8C%80%EC%88%98" title="외대수">외대수</a></li> <li><a href="/w/index.php?title=%EC%9D%B4%EC%A4%91%EB%B2%A1%ED%84%B0&action=edit&redlink=1" class="new" title="이중벡터 (없는 문서)">이중벡터</a></li> <li><a href="/w/index.php?title=%EB%8B%A4%EC%A4%91%EB%B2%A1%ED%84%B0&action=edit&redlink=1" class="new" title="다중벡터 (없는 문서)">다중벡터</a></li> <li><a href="/wiki/%ED%85%90%EC%84%9C" title="텐서">텐서</a></li> <li><a href="/w/index.php?title=%EC%95%84%EC%9A%B0%ED%84%B0%EB%AA%A8%ED%94%BC%EC%A6%98&action=edit&redlink=1" class="new" title="아우터모피즘 (없는 문서)">아우터모피즘</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%ED%96%89%EB%A0%AC" title="행렬">행렬</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EB%B8%94%EB%A1%9D_%ED%96%89%EB%A0%AC" title="블록 행렬">블록 행렬</a></li> <li><a href="/wiki/%ED%96%89%EB%A0%AC_%EB%B6%84%ED%95%B4" title="행렬 분해">행렬 분해</a></li> <li><a href="/wiki/%EA%B0%80%EC%97%AD%ED%96%89%EB%A0%AC" title="가역행렬">가역행렬</a></li> <li><a href="/wiki/%EC%86%8C%ED%96%89%EB%A0%AC%EC%8B%9D" class="mw-redirect" title="소행렬식">소행렬식</a></li> <li><a href="/wiki/%ED%96%89%EB%A0%AC_%EA%B3%B1%EC%85%88" title="행렬 곱셈">행렬 곱셈</a></li> <li><a href="/wiki/%EA%B3%84%EC%88%98_(%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99)" title="계수 (선형대수학)">계수</a></li> <li><a href="/wiki/%EB%B3%80%ED%99%98%ED%96%89%EB%A0%AC" title="변환행렬">변환행렬</a></li> <li><a href="/wiki/%ED%81%AC%EB%9D%BC%EB%A9%94%EB%A5%B4_%EA%B3%B5%EC%8B%9D" class="mw-redirect" title="크라메르 공식">크라메르 공식</a></li> <li><a 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href="/wiki/%ED%85%90%EC%84%9C%EA%B3%B1" title="텐서곱">텐서곱</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%EC%88%98%EC%B9%98%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99" title="수치선형대수학">수치선형대수학</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EB%B6%80%EB%8F%99%EC%86%8C%EC%88%98%EC%A0%90" title="부동소수점">부동소수점</a></li> <li><a href="/w/index.php?title=%EC%88%98%EC%B9%98%EC%A0%81_%EC%95%88%EC%A0%95%EC%84%B1&action=edit&redlink=1" class="new" title="수치적 안정성 (없는 문서)">수치적 안정성</a></li> <li><a href="/w/index.php?title=BLAS&action=edit&redlink=1" class="new" title="BLAS (없는 문서)">BLAS</a>(Basic Linear Algebra Subprogram)</li> <li><a href="/wiki/%ED%9D%AC%EC%86%8C%ED%96%89%EB%A0%AC" class="mw-redirect" title="희소행렬">희소행렬</a></li> <li><a href="/w/index.php?title=%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99_%EB%9D%BC%EC%9D%B4%EB%B8%8C%EB%9F%AC%EB%A6%AC_%EB%B9%84%EA%B5%90&action=edit&redlink=1" class="new" title="선형대수학 라이브러리 비교 (없는 문서)">선형대수학 라이브러리 비교</a></li> <li><a href="/w/index.php?title=%EC%88%98%EC%B9%98_%EB%B6%84%EC%84%9D_%EC%86%8C%ED%94%84%ED%8A%B8%EC%9B%A8%EC%96%B4_%EB%B9%84%EA%B5%90&action=edit&redlink=1" class="new" title="수치 분석 소프트웨어 비교 (없는 문서)">수치 분석 소프트웨어 비교</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3" style="font-weight:bold;"><div> <ul><li><span typeof="mw:File"><span title="분류"><img alt="분류" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Folder_Hexagonal_Icon.svg/16px-Folder_Hexagonal_Icon.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Folder_Hexagonal_Icon.svg/24px-Folder_Hexagonal_Icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Folder_Hexagonal_Icon.svg/32px-Folder_Hexagonal_Icon.svg.png 2x" data-file-width="36" data-file-height="31" /></span></span> <a href="/wiki/%EB%B6%84%EB%A5%98:%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99" title="분류:선형대수학">분류</a></li> <li><span typeof="mw:File"><span title="목록 문서"><img alt="목록 문서" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/w/index.php?title=%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99_%EC%A3%BC%EC%A0%9C_%EB%AA%A9%EB%A1%9D&action=edit&redlink=1" class="new" title="선형대수학 주제 목록 (없는 문서)">개요</a></li> <li><span typeof="mw:File"><span title="포털"><img alt="포털" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Portal.svg/16px-Portal.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Portal.svg/24px-Portal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Portal.svg/32px-Portal.svg.png 2x" data-file-width="36" data-file-height="32" /></span></span> <a href="/wiki/%ED%8F%AC%ED%84%B8:%EC%88%98%ED%95%99" title="포털:수학">수학 포털</a></li> <li><span typeof="mw:File"><span title="위키책"><img alt="위키책" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/16px-Wikibooks-logo.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/24px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/32px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></span></span> <a href="https://en.wikipedia.org/wiki/wikibooks:Linear_algebra" class="extiw" title="en:wikibooks:Linear algebra">위키책</a></li> <li><span typeof="mw:File"><span title="위키배움터"><img alt="위키배움터" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/16px-Wikiversity-logo.svg.png" decoding="async" width="16" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/24px-Wikiversity-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/32px-Wikiversity-logo.svg.png 2x" data-file-width="1000" data-file-height="800" /></span></span> <a href="https://en.wikipedia.org/wiki/wikiversity:Linear_algebra" class="extiw" title="en:wikiversity:Linear algebra">위키배움터</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" 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href="/wiki/%EB%B6%84%EB%A5%98:Harv_%EB%B0%8F_Sfn%EC%97%90_%EB%8C%80%EC%83%81%EC%9D%B4_%EC%97%86%EB%8A%94_%EC%98%A4%EB%A5%98%EA%B0%80_%EC%9E%88%EB%8A%94_%EB%AC%B8%EC%84%9C" title="분류:Harv 및 Sfn에 대상이 없는 오류가 있는 문서">Harv 및 Sfn에 대상이 없는 오류가 있는 문서</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EC%9C%84%ED%82%A4%EB%8D%B0%EC%9D%B4%ED%84%B0_%EC%86%8D%EC%84%B1_P7859%EB%A5%BC_%EC%82%AC%EC%9A%A9%ED%95%98%EB%8A%94_%EB%AC%B8%EC%84%9C" title="분류:위키데이터 속성 P7859를 사용하는 문서">위키데이터 속성 P7859를 사용하는 문서</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EC%98%81%EC%96%B4_%ED%91%9C%EA%B8%B0%EB%A5%BC_%ED%8F%AC%ED%95%A8%ED%95%9C_%EB%AC%B8%EC%84%9C" title="분류:영어 표기를 포함한 문서">영어 표기를 포함한 문서</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> 이 문서는 2023년 2월 8일 (수) 17:41에 마지막으로 편집되었습니다.</li> <li id="footer-info-copyright">모든 문서는 <a rel="nofollow" class="external text" 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