@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr: a skos:ConceptScheme .
psr:-WHDHQH7N-Q
  skos:prefLabel "algèbre multilinéaire"@fr, "multilinear algebra"@en ;
  a skos:Concept ;
  skos:narrower psr:-PHZS8DDC-8 .

psr:-VTR5XXB2-M
  skos:prefLabel "identité"@fr, "identity"@en ;
  a skos:Concept ;
  skos:narrower psr:-PHZS8DDC-8 .

psr:-PHZS8DDC-8
  skos:broader psr:-VTR5XXB2-M, psr:-WHDHQH7N-Q ;
  skos:prefLabel "Lagrange's identity"@en, "identité de Lagrange"@fr ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Identit%C3%A9_de_Lagrange>, <https://en.wikipedia.org/wiki/Lagrange%27s_identity> ;
  skos:definition """In algebra, <b>Lagrange's identity</b>, named after Joseph Louis Lagrange, is:
<br/><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\egin{aligned}\\\\left(\\\\sum _{k=1}^{n}a_{k}^{2}\\
ight)\\\\left(\\\\sum _{k=1}^{n}b_{k}^{2}\\
ight)-\\\\left(\\\\sum _{k=1}^{n}a_{k}b_{k}\\
ight)^{2}&amp;=\\\\sum _{i=1}^{n-1}\\\\sum _{j=i+1}^{n}\\\\left(a_{i}b_{j}-a_{j}b_{i}\\
ight)^{2}\\\\\\\\&amp;\\\\left(={\\rac {1}{2}}\\\\sum _{i=1}^{n}\\\\sum _{j=1,j\\
eq i}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}\\
ight),\\\\end{aligned}}}">
<br/>  <semantics>
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<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\egin{aligned}\\\\left(\\\\sum _{k=1}^{n}a_{k}^{2}\\
ight)\\\\left(\\\\sum _{k=1}^{n}b_{k}^{2}\\
ight)-\\\\left(\\\\sum _{k=1}^{n}a_{k}b_{k}\\
ight)^{2}&amp;=\\\\sum _{i=1}^{n-1}\\\\sum _{j=i+1}^{n}\\\\left(a_{i}b_{j}-a_{j}b_{i}\\
ight)^{2}\\\\\\\\&amp;\\\\left(={\\rac {1}{2}}\\\\sum _{i=1}^{n}\\\\sum _{j=1,j\\
eq i}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}\\
ight),\\\\end{aligned}}}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b2913f2fc530bda2a5d40ded2a9d6b30a1aad5c" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -7.505ex; width:68.746ex; height:16.176ex;" alt="{\\\\displaystyle {\\egin{aligned}\\\\left(\\\\sum _{k=1}^{n}a_{k}^{2}\\
ight)\\\\left(\\\\sum _{k=1}^{n}b_{k}^{2}\\
ight)-\\\\left(\\\\sum _{k=1}^{n}a_{k}b_{k}\\
ight)^{2}&amp;=\\\\sum _{i=1}^{n-1}\\\\sum _{j=i+1}^{n}\\\\left(a_{i}b_{j}-a_{j}b_{i}\\
ight)^{2}\\\\\\\\&amp;\\\\left(={\\rac {1}{2}}\\\\sum _{i=1}^{n}\\\\sum _{j=1,j\\
eq i}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}\\
ight),\\\\end{aligned}}}"></div>
<br/>which applies to any two sets {<i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, ..., <i>a<sub>n</sub></i>} and {<i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, ..., <i>b<sub>n</sub></i>} of real or complex numbers (or more generally, elements of a commutative ring). This identity is a generalisation of the Brahmagupta–Fibonacci identity and a special form of the Binet–Cauchy identity. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Lagrange%27s_identity">https://en.wikipedia.org/wiki/Lagrange%27s_identity</a>)"""@en, """En mathématiques, et plus particulièrement en algèbre, l'identité de Lagrange, découverte par Joseph Louis Lagrange, est une formule transformant un produit de sommes de carrés en une autre somme de carrés ; elle a d'importantes conséquences sur les propriétés du produit vectoriel. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Identit%C3%A9_de_Lagrange">https://fr.wikipedia.org/wiki/Identit%C3%A9_de_Lagrange</a>)"""@fr ;
  dc:modified "2023-07-13"^^xsd:date ;
  a skos:Concept ;
  dc:created "2023-07-13"^^xsd:date ;
  skos:inScheme psr: .