@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:-HLPMWTJF-1
  skos:exactMatch <https://en.wikipedia.org/wiki/Projection_(linear_algebra)>, <https://fr.wikipedia.org/wiki/Projecteur_(math%C3%A9matiques)> ;
  skos:altLabel "projecteur"@fr ;
  skos:prefLabel "projection"@fr, "projection"@en ;
  skos:definition """En algèbre linéaire, un <b>projecteur</b> (ou une <b>projection</b>) est une application linéaire qu'on peut présenter de deux façons équivalentes :  <ul><li>une projection linéaire associée à  une décomposition d'un espace vectoriel <i>E</i> comme somme de deux sous-espaces supplémentaires, c'est-à-dire qu'elle permet d'obtenir un des termes de la décomposition correspondante ;</li> <li>une application linéaire idempotente : elle vérifie <span class="texhtml"><i>p</i><sup>2</sup> = <i>p</i></span>.</li></ul> Dans un espace hilbertien ou même seulement préhilbertien, une projection pour laquelle les deux supplémentaires sont orthogonaux est appelée projection orthogonale. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Projecteur_(math%C3%A9matiques)">https://fr.wikipedia.org/wiki/Projecteur_(math%C3%A9matiques)</a>)"""@fr, """In linear algebra and functional analysis, a <b>projection</b> is a linear transformation <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 P}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>P</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle P}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\\\\displaystyle P}"></span> from a vector space to itself (an endomorphism) such that <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 P\\\\circ P=P}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>P</mi>         <mo>∘<!-- ∘ --></mo>         <mi>P</mi>         <mo>=</mo>         <mi>P</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle P\\\\circ P=P}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5369d895625034bc50c9f28975e3293ef6f2105b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.529ex; height:2.176ex;" alt="{\\\\displaystyle P\\\\circ P=P}"></span>. That is, whenever <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 P}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>P</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle P}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\\\\displaystyle P}"></span> is applied twice to any vector, it gives the same result as if it were applied once (i.e. <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 P}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>P</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle P}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\\\\displaystyle P}"></span> is idempotent). It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection.  One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.  
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Projection_(linear_algebra)">https://en.wikipedia.org/wiki/Projection_(linear_algebra)</a>)"""@en ;
  a skos:Concept ;
  skos:broader psr:-VVTJ8P47-K ;
  skos:narrower psr:-DTDHZQD0-9 ;
  skos:inScheme psr: ;
  dc:modified "2024-10-18"^^xsd:date .

psr: a skos:ConceptScheme .
psr:-DTDHZQD0-9
  skos:prefLabel "orthogonal projection"@en, "projection orthogonale"@fr ;
  a skos:Concept ;
  skos:broader psr:-HLPMWTJF-1 .

psr:-VVTJ8P47-K
  skos:prefLabel "application linéaire"@fr, "linear map"@en ;
  a skos:Concept ;
  skos:narrower psr:-HLPMWTJF-1 .