@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr:-S6WX2HJH-P
  skos:prefLabel "application bilinéaire"@fr, "bilinear map"@en ;
  a skos:Concept ;
  skos:broader psr:-S7M9JFHC-2 .

psr:-Q8X6082L-Q
  skos:prefLabel "forme multilinéaire"@fr, "multilinear form"@en ;
  a skos:Concept ;
  skos:broader psr:-S7M9JFHC-2 .

psr:-S7M9JFHC-2
  skos:inScheme psr: ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Application_multilin%C3%A9aire>, <https://en.wikipedia.org/wiki/Multilinear_map> ;
  skos:prefLabel "multilinear map"@en, "application multilinéaire"@fr ;
  skos:altLabel "multilinear mapping"@en ;
  skos:narrower psr:-S6WX2HJH-P, psr:-Q8X6082L-Q ;
  a skos:Concept ;
  skos:definition """En algèbre linéaire, une application multilinéaire est une application à plusieurs variables vectorielles et à valeurs vectorielles qui est linéaire en chaque variable. Une application multilinéaire à valeurs scalaires est appelée forme multilinéaire. Une application multilinéaire à deux variables vectorielles est dite bilinéaire.
<br/>Quelques exemples classiques :
<br/>- le produit scalaire est une forme bilinéaire symétrique;
<br/>- le déterminant est une forme multilinéaire antisymétrique des colonnes (ou lignes) d'une matrice carrée. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Application_multilin%C3%A9aire">https://fr.wikipedia.org/wiki/Application_multilin%C3%A9aire</a>)"""@fr, """In linear algebra, a <b>multilinear map</b> is a function of several variables that is linear separately in each variable.  More precisely, a multilinear map is a function
<br/>
<br/><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 f\\\\colon V_{1}\\	imes \\\\cdots \\	imes V_{n}\\	o W{\\	ext{,}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mo>:<!-- : --></mo>
<br/>        <msub>
<br/>          <mi>V</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>×<!-- × --></mo>
<br/>        <mo>⋯<!-- ⋯ --></mo>
<br/>        <mo>×<!-- × --></mo>
<br/>        <msub>
<br/>          <mi>V</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">→<!-- → --></mo>
<br/>        <mi>W</mi>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mtext>,</mtext>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f\\\\colon V_{1}\\	imes \\\\cdots \\	imes V_{n}\\	o W{\\	ext{,}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a791997e08e880ad7aeb9808d71b559be172eeb4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:22.396ex; height:2.509ex;" alt="f\\\\colon V_{1}\\	imes \\\\cdots \\	imes V_{n}\\	o W{\\	ext{,}}"></span></dd></dl>
<br/>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle V_{1},\\\\ldots ,V_{n}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>V</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>,</mo>
<br/>        <mo>…<!-- … --></mo>
<br/>        <mo>,</mo>
<br/>        <msub>
<br/>          <mi>V</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle V_{1},\\\\ldots ,V_{n}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/056aea51dd62880a078cf84fe69ff7d09afb20b2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:10.161ex; height:2.509ex;" alt="V_{1},\\\\ldots ,V_{n}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle W}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>W</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle W}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="W"></span> are vector spaces (or modules over a commutative ring), with the following property: for each <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 i}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>i</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle i}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="i"></span>, if all of the variables but <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle v_{i}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>v</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle v_{i}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dffe5726650f6daac54829972a94f38eb8ec127" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="v_{i}"></span> are held constant, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f(v_{1},\\\\ldots ,v_{i},\\\\ldots ,v_{n})}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <msub>
<br/>          <mi>v</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>,</mo>
<br/>        <mo>…<!-- … --></mo>
<br/>        <mo>,</mo>
<br/>        <msub>
<br/>          <mi>v</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>,</mo>
<br/>        <mo>…<!-- … --></mo>
<br/>        <mo>,</mo>
<br/>        <msub>
<br/>          <mi>v</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f(v_{1},\\\\ldots ,v_{i},\\\\ldots ,v_{n})}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c39026502b25d6b385c4fde827d545482b76e073" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:19.9ex; height:2.843ex;" alt="{\\\\displaystyle f(v_{1},\\\\ldots ,v_{i},\\\\ldots ,v_{n})}"></span> is a linear function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle v_{i}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>v</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle v_{i}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dffe5726650f6daac54829972a94f38eb8ec127" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="v_{i}"></span>.
<br/> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Multilinear_map">https://en.wikipedia.org/wiki/Multilinear_map</a>)"""@en ;
  skos:broader psr:-WHDHQH7N-Q .

psr: a skos:ConceptScheme .
psr:-WHDHQH7N-Q
  skos:prefLabel "algèbre multilinéaire"@fr, "multilinear algebra"@en ;
  a skos:Concept ;
  skos:narrower psr:-S7M9JFHC-2 .