GermanWikipedia - /page/entry.mu

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Multilinearform</h1>

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-Multilinearform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> ist in der Mathematik eine <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</a>, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Argumenten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}\in V_{i},\;i\in \{1,\ldots ,p\}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{i}\in V_{i},\;i\in \{1,\ldots ,p\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1be31966da930b1d3502dfbebc690c795a4973fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.08ex; height:2.843ex;" alt="{\displaystyle v_{i}\in V_{i},\;i\in \{1,\ldots ,p\}}" loading="lazy"> aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">-<a href="Vektorraum" title="Vektorraum">Vektorräumen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{1},\ldots ,V_{p}}">

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<annotation encoding="application/x-tex">{\displaystyle V_{1},\ldots ,V_{p}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35946e2f9af1c0989447f99e90e8d5fc9866d545.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.002ex; height:2.843ex;" alt="{\displaystyle V_{1},\ldots ,V_{p}}" loading="lazy"> einen Wert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (v_{1},\ldots ,v_{p})\in K}">

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<annotation encoding="application/x-tex">{\displaystyle \omega (v_{1},\ldots ,v_{p})\in K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1356f6e8f6c42c3ba96db3508e91b1ad42b4437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.709ex; height:3.009ex;" alt="{\displaystyle \omega (v_{1},\ldots ,v_{p})\in K}" loading="lazy"> zuordnet und in jeder Komponente linear ist. Im allgemeineren Fall, dass der Bildraum selbst ein Vektorraum ist, oder Bild- und Zielräume <a href="Modul_(Mathematik)" title="Modul (Mathematik)">Moduln</a> sind, spricht man von einer <a href="Multilineare_Abbildung" title="Multilineare Abbildung">multilinearen Abbildung</a>.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Eine Abbildung

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\omega :\ V_{1}\times \cdots \times V_{p}&\rightarrow K\\(v_{1},\ldots ,v_{p})\ &\mapsto \omega \left(v_{1},\dots ,v_{p}\right)\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/301bd1697ab69ca699085334586dbce14a0ef08c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.746ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\omega :\ V_{1}\times \cdots \times V_{p}&\rightarrow K\\(v_{1},\ldots ,v_{p})\ &\mapsto \omega \left(v_{1},\dots ,v_{p}\right)\end{aligned}}}" loading="lazy"></dd></dl>

heißt Multilinearform, wenn für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{j}\in V_{j},j\in \{1,\ldots ,p\}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{j}\in V_{j},j\in \{1,\ldots ,p\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d45d82018d75d183d77951e7b8238f35d1cfde0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.811ex; height:3.009ex;" alt="{\displaystyle v_{j}\in V_{j},j\in \{1,\ldots ,p\}}" loading="lazy"> und alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \{1,\ldots ,p\}}">

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<annotation encoding="application/x-tex">{\displaystyle i\in \{1,\ldots ,p\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85a50a95991f624cd213baa24b3c4a157a9fcc5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.478ex; height:2.843ex;" alt="{\displaystyle i\in \{1,\ldots ,p\}}" loading="lazy"> folgende zwei Bedingungen erfüllt sind:

Für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in K}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda \in K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22d95fcd9b07168a8162820f7fab4d8ee43366e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.262ex; height:2.176ex;" alt="{\displaystyle \lambda \in K}" loading="lazy"> gilt

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<annotation encoding="application/x-tex">{\displaystyle \omega \left(v_{1},\ldots ,\lambda \;v_{i},\ldots ,v_{p}\right)=\lambda \;\omega \left(v_{1},\ldots ,v_{i},\ldots ,v_{p}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4694379fbba5336bd1db0905d3675c085ff9613e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.688ex; height:3.009ex;" alt="{\displaystyle \omega \left(v_{1},\ldots ,\lambda \;v_{i},\ldots ,v_{p}\right)=\lambda \;\omega \left(v_{1},\ldots ,v_{i},\ldots ,v_{p}\right)}" loading="lazy"></dd></dl>

und für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w\in V_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle w\in V_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53ca747b37206e03df01e24c3c3b46929fbd8436.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.66ex; height:2.509ex;" alt="{\displaystyle w\in V_{i}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \omega \left(v_{1},\ldots ,v_{i}+w,\ldots ,v_{p}\right)=\omega \left(v_{1},\ldots ,v_{i},\ldots ,v_{p}\right)+\omega \left(v_{1},\ldots ,w,\ldots ,v_{p}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e509154e0c499479d0361e9284303d66a2d83d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:71.064ex; height:3.009ex;" alt="{\displaystyle \omega \left(v_{1},\ldots ,v_{i}+w,\ldots ,v_{p}\right)=\omega \left(v_{1},\ldots ,v_{i},\ldots ,v_{p}\right)+\omega \left(v_{1},\ldots ,w,\ldots ,v_{p}\right)}" loading="lazy">.</dd></dl>

Die Menge aller multilinearen Abbildungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {J}}^{p}(V_{1},\ldots ,V_{p})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {J}}^{p}(V_{1},\ldots ,V_{p})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/976e52900b1bdc9ffba568571515ac6e7decd4f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.933ex; height:3.009ex;" alt="{\displaystyle {\mathcal {J}}^{p}(V_{1},\ldots ,V_{p})}" loading="lazy"> bildet einen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">-Vektorraum. Im Fall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{1}=\cdots =V_{p}=:V}">

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<annotation encoding="application/x-tex">{\displaystyle V_{1}=\cdots =V_{p}=:V}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ccdc8cc7bd604d4338e932dda9c82004a2d10f56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.277ex; height:2.843ex;" alt="{\displaystyle V_{1}=\cdots =V_{p}=:V}" loading="lazy"> schreibt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {J}}^{p}(V):={\mathcal {J}}^{p}(V,\ldots ,V)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {J}}^{p}(V):={\mathcal {J}}^{p}(V,\ldots ,V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38bbf377a2bd41d21bdfddd413b24e7c3adc68aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.146ex; height:2.843ex;" alt="{\displaystyle {\mathcal {J}}^{p}(V):={\mathcal {J}}^{p}(V,\ldots ,V)}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Alternierende_Multilinearformen">Alternierende Multilinearformen</h2></div>

Eine Multilinearform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in {\mathcal {J}}^{p}(V)}">

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<annotation encoding="application/x-tex">{\displaystyle \omega \in {\mathcal {J}}^{p}(V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1928e4bdd40fde5c11e4e0c968cebb186b44a6ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.004ex; height:2.843ex;" alt="{\displaystyle \omega \in {\mathcal {J}}^{p}(V)}" loading="lazy"> heißt alternierend, falls sie null ergibt, wenn zweimal derselbe Vektor eingesetzt wird, d. h.

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<annotation encoding="application/x-tex">{\displaystyle \omega \left(\dots ,v,\dots ,v,\dots \right)=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/abba3b9e917d0b0999f7d548665ddd08c8ea07fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.238ex; height:2.843ex;" alt="{\displaystyle \omega \left(\dots ,v,\dots ,v,\dots \right)=0}" loading="lazy"></dd></dl>

für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in V}">

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<annotation encoding="application/x-tex">{\displaystyle v\in V}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.756ex; height:2.176ex;" alt="{\displaystyle v\in V}" loading="lazy">.<a href="#cite_note-eom-1">[1]</a>

In diesem Fall folgt auch, dass die Form schiefsymmetrisch ist, das heißt, dass sie bei Vertauschung von zwei beliebigen Argumenten ihr <a href="Vorzeichen_(Zahl)" title="Vorzeichen (Zahl)">Vorzeichen</a> wechselt, also

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<annotation encoding="application/x-tex">{\displaystyle \omega \left(v_{1},\dots ,v_{i},\dots ,v_{j},\dots ,v_{p}\right)=-\omega \left(v_{1},\dots ,v_{j},\dots ,v_{i},\dots ,v_{p}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52c8a6edd44501ecf2a2f2f7fce252a191f054b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:59.927ex; height:3.009ex;" alt="{\displaystyle \omega \left(v_{1},\dots ,v_{i},\dots ,v_{j},\dots ,v_{p}\right)=-\omega \left(v_{1},\dots ,v_{j},\dots ,v_{i},\dots ,v_{p}\right)}" loading="lazy"></dd></dl>

für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}\in V,\;k\in \{1,\ldots ,p\}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{k}\in V,\;k\in \{1,\ldots ,p\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76530f6b783d21dd9040b86ba88c379fefe7ef08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.41ex; height:2.843ex;" alt="{\displaystyle v_{k}\in V,\;k\in \{1,\ldots ,p\}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j\in \{1,\ldots ,p\},\;i\neq j}">

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<annotation encoding="application/x-tex">{\displaystyle i,j\in \{1,\ldots ,p\},\;i\neq j}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05e221acb327cb84a3c9eaacab6abc513257215c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.008ex; height:2.843ex;" alt="{\displaystyle i,j\in \{1,\ldots ,p\},\;i\neq j}" loading="lazy">. Die umgekehrte Implikation – dass alle schiefsymmetrischen Multilinearformen alternierend sind – gilt aber nur, wenn die <a href="Charakteristik_(Algebra)" title="Charakteristik (Algebra)">Charakteristik</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6419d3aa99701ca996737b17a5e1174d53e6c9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.842ex; height:2.176ex;" alt="{\displaystyle K=\mathbb {R} }" loading="lazy">.<a href="#cite_note-eom-1">[1]</a>

Ist allgemeiner <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \in S_{p}}">

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<annotation encoding="application/x-tex">{\displaystyle \pi \in S_{p}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/424a4a4237c6874745ebabed911b1a8e04b609ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.657ex; height:2.843ex;" alt="{\displaystyle \pi \in S_{p}}" loading="lazy"> eine beliebige <a href="Permutation" title="Permutation">Permutation</a> der Indizes, dann gilt

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<annotation encoding="application/x-tex">{\displaystyle \omega \left(v_{\pi (1)},\dotsc ,v_{\pi (p)}\right)=\operatorname {sign} (\pi )\cdot \omega \left(v_{1},\dotsc ,v_{p}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c3305cf65b9a17a67538daf6eeabe1b7872ebd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:43.078ex; height:3.343ex;" alt="{\displaystyle \omega \left(v_{\pi (1)},\dotsc ,v_{\pi (p)}\right)=\operatorname {sign} (\pi )\cdot \omega \left(v_{1},\dotsc ,v_{p}\right)}" loading="lazy">,</dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sign} (\pi )}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ae4d9c3bc2ae8cddb7d99a0096788d21d1ba1a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.159ex; height:2.843ex;" alt="{\displaystyle \operatorname {sign} (\pi )}" loading="lazy"> das <a href="Vorzeichen_(Permutation)" title="Vorzeichen (Permutation)">Signum</a> der Permutation bezeichnet.

Die Menge aller alternierenden Multilinearformen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ^{p}(V)}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65b2477574743f273cf13bb6897cc92d0c2a856c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.334ex; height:2.843ex;" alt="{\displaystyle \Omega ^{p}(V)}" loading="lazy"> ist ein <a href="Untervektorraum" title="Untervektorraum">Untervektorraum</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {J}}^{p}(V)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {J}}^{p}(V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d6fa441f399badd1bd5444555de8f15bc1bbb1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.718ex; height:2.843ex;" alt="{\displaystyle {\mathcal {J}}^{p}(V)}" loading="lazy">. Wichtig ist der Spezialfall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ p=\dim V}">

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<annotation encoding="application/x-tex">{\displaystyle \ p=\dim V}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00c8d332f7b7d8fe19cac14b8971c20d0afcbbc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.898ex; height:2.509ex;" alt="{\displaystyle \ p=\dim V}" loading="lazy">. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ^{p}(V)}">

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<annotation encoding="application/x-tex">{\displaystyle \Omega ^{p}(V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65b2477574743f273cf13bb6897cc92d0c2a856c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.334ex; height:2.843ex;" alt="{\displaystyle \Omega ^{p}(V)}" loading="lazy"> ein eindimensionaler Unterraum von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {J}}^{p}(V)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {J}}^{p}(V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d6fa441f399badd1bd5444555de8f15bc1bbb1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.718ex; height:2.843ex;" alt="{\displaystyle {\mathcal {J}}^{p}(V)}" loading="lazy">, und seine Elemente heißen <a href="Determinantenfunktion" title="Determinantenfunktion">Determinantenfunktionen</a>.

Auf dem durch alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ^{p}(V),p=0,1,2,\ldots }">

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<annotation encoding="application/x-tex">{\displaystyle \Omega ^{p}(V),p=0,1,2,\ldots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c866bd4f46ac0971dff6fc6f2c0c0b869addfbe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.948ex; height:2.843ex;" alt="{\displaystyle \Omega ^{p}(V),p=0,1,2,\ldots }" loading="lazy"> erzeugten Vektorraum lässt sich die Struktur einer <a href="Algebra_%C3%BCber_einem_K%C3%B6rper" title="Algebra über einem Körper">Algebra</a> definieren. Diese Algebra heißt <a href="Gra%C3%9Fmann-Algebra" title="Graßmann-Algebra">Graßmann-Algebra</a>.

<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

<ol><li><a href="Linearform" title="Linearform">Linearformen</a> sind genau die 1-Multilinearformen.</li>

<li><a href="Bilinearform" title="Bilinearform">Bilinearformen</a> sind genau die 2-Multilinearformen. Antisymmetrische Bilinearformen sind alternierende Multilinearformen (wenn die Charakteristik von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> nicht 2 ist).</li>

<li>Bildet man aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> Vektoren durch Zusammenfassen eine quadratische <a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a>, so ist die <a href="Determinante_(Mathematik)" class="mw-redirect" title="Determinante (Mathematik)">Determinante</a> dieser Matrix eine alternierende, normierte Multilinearform. Im dreidimensionalen Fall ist also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> definiert durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \left(v_{1},v_{2},v_{3}\right):=\det {\begin{pmatrix}v_{1x}&v_{2x}&v_{3x}\\v_{1y}&v_{2y}&v_{3y}\\v_{1z}&v_{2z}&v_{3z}\end{pmatrix}}}">

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \omega \left(v_{1},v_{2},v_{3}\right):=\det {\begin{pmatrix}v_{1x}&v_{2x}&v_{3x}\\v_{1y}&v_{2y}&v_{3y}\\v_{1z}&v_{2z}&v_{3z}\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/256da31add8fbff950c31b196260d4ced4447f0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:38.447ex; height:9.509ex;" alt="{\displaystyle \omega \left(v_{1},v_{2},v_{3}\right):=\det {\begin{pmatrix}v_{1x}&v_{2x}&v_{3x}\\v_{1y}&v_{2y}&v_{3y}\\v_{1z}&v_{2z}&v_{3z}\end{pmatrix}}}" loading="lazy"> eine alternierende 3-Multilinearform. Dabei sind die Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1},v_{2},v_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{1},v_{2},v_{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12e5ac2b8e42965e3d97dbe06d1031361bc310e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.613ex; height:2.009ex;" alt="{\displaystyle v_{1},v_{2},v_{3}}" loading="lazy"> folgendermaßen in Koordinaten dargestellt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1}={\begin{pmatrix}v_{1x}\\v_{1y}\\v_{1z}\end{pmatrix}},\quad \quad v_{2}={\begin{pmatrix}v_{2x}\\v_{2y}\\v_{2z}\end{pmatrix}},\quad \quad v_{3}={\begin{pmatrix}v_{3x}\\v_{3y}\\v_{3z}\end{pmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{1}={\begin{pmatrix}v_{1x}\\v_{1y}\\v_{1z}\end{pmatrix}},\quad \quad v_{2}={\begin{pmatrix}v_{2x}\\v_{2y}\\v_{2z}\end{pmatrix}},\quad \quad v_{3}={\begin{pmatrix}v_{3x}\\v_{3y}\\v_{3z}\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/835124eecd5a5354a348848b19f6ae6b8698f21d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:51.02ex; height:9.509ex;" alt="{\displaystyle v_{1}={\begin{pmatrix}v_{1x}\\v_{1y}\\v_{1z}\end{pmatrix}},\quad \quad v_{2}={\begin{pmatrix}v_{2x}\\v_{2y}\\v_{2z}\end{pmatrix}},\quad \quad v_{3}={\begin{pmatrix}v_{3x}\\v_{3y}\\v_{3z}\end{pmatrix}}}" loading="lazy">.</li>

<li>Kovariante <a href="Tensor" title="Tensor">Tensoren</a> sind Multilinearformen: In dem Fall, dass alle Vektorräume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle V_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f300b83673e961a9d48f3862216b167f94e5668c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle V_{i}}" loading="lazy"> identisch sind (also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}=V}">

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{i}=V}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3edf760a9e75d5833cfe9992db1772e60ce19a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.041ex; height:2.509ex;" alt="{\displaystyle V_{i}=V}" loading="lazy">), ist die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-Multilinearform auch ein kovarianter Tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-ter Stufe. Im selben Fall sind die alternierenden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-Multilinearformen auch total antisymmetrische Tensoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-ter Stufe.</li>

<li>Eine <a href="Differentialform" title="Differentialform">Differentialform</a> ordnet einem Punkt einer <a href="Mannigfaltigkeit" title="Mannigfaltigkeit">Mannigfaltigkeit</a> eine alternierende Multilinearform auf dem zugehörigen <a href="Tangentialraum" title="Tangentialraum">Tangentialraum</a> zu.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Gerd_Fischer_(Mathematiker)" title="Gerd Fischer (Mathematiker)">Gerd Fischer</a>: Lineare Algebra. Vieweg-Verlag, ISBN 3-528-03217-0.</li>

<li><a href="Hans-Joachim_Kowalsky" title="Hans-Joachim Kowalsky">Hans-Joachim Kowalsky</a>, <a href="Gerhard_O._Michler" title="Gerhard O. Michler">Gerhard O. Michler</a>: Lineare Algebra. De Gruyter, Berlin 2003, ISBN 978-3-11-017963-7.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-eom-1">↑ <a href="#cite_ref-eom_1-0">a</a> <a href="#cite_ref-eom_1-1">b</a> <a href="Arkadi_Lwowitsch_Onischik" class="mw-redirect" title="Arkadi Lwowitsch Onischik">Arkady L'vovich Onishchik</a>: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Multilinear mapping</cite>. In: <a href="Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a> (Hrsg.): <cite class="lang" lang="en" dir="auto" style="font-style:italic"><a href="Encyclopedia_of_Mathematics" class="mw-redirect" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></cite>. Springer-Verlag und <a href="European_Mathematical_Society" title="European Mathematical Society">EMS</a> Press, Berlin 2002, ISBN 1-55608-010-7 (englisch, <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/multilinear_mapping">encyclopediaofmath.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Multilinearform&rft.atitle=Multilinear+mapping&rft.au=Arkady+L%27vovich+Onishchik&rft.btitle=Encyclopedia+of+Mathematics&rft.date=2002&rft.genre=book&rft.isbn=1556080107&rft.place=Berlin&rft.pub=Springer-Verlag+und+EMS+Press" style="display:none">

</li>

</ol></div><div><div class="zim-footer">

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