@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:-W0JJX1W8-X
  skos:prefLabel "vector space"@en, "espace vectoriel"@fr ;
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psr: a skos:ConceptScheme .
psr:-Q8T0JSP7-0
  skos:broader psr:-W0JJX1W8-X, psr:-M3NJVVTK-V ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Drapeau_(math%C3%A9matiques)>, <https://en.wikipedia.org/wiki/Flag_(linear_algebra)> ;
  a skos:Concept ;
  dc:created "2023-08-31"^^xsd:date ;
  skos:inScheme psr: ;
  dc:modified "2023-08-31"^^xsd:date ;
  skos:definition """In mathematics, particularly in linear algebra, a <b>flag</b> is an increasing sequence of subspaces of a finite-dimensional vector space <i>V</i>. Here "increasing" means each is a proper subspace of the next (see filtration):
<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 \\\\{0\\\\}=V_{0}\\\\subset V_{1}\\\\subset V_{2}\\\\subset \\\\cdots \\\\subset V_{k}=V.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
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<br/>        <mo fence="false" stretchy="false">{</mo>
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<br/>          <mi>V</mi>
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<br/>          <mi>V</mi>
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<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\{0\\\\}=V_{0}\\\\subset V_{1}\\\\subset V_{2}\\\\subset \\\\cdots \\\\subset V_{k}=V.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b61cfe3efa97cdfae17e85be311fbe9398aa20b4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:36.908ex; height:2.843ex;" alt="\\\\{0\\\\}=V_{0}\\\\subset V_{1}\\\\subset V_{2}\\\\subset \\\\cdots \\\\subset V_{k}=V."></span></dd></dl>
<br/>The term <i>flag</i> is motivated by a particular example resembling a flag: the zero point, a line, and a plane correspond to a nail, a staff, and a sheet of fabric.
<br/>If we write that dim<i>V</i><sub><i>i</i></sub> = <i>d</i><sub><i>i</i></sub> then we have
<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 0=d_{0}<d_{1}<d_{2}<\\\\cdots <d_{k}=n,}">
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<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle 0=d_{0}&lt;d_{1}&lt;d_{2}&lt;\\\\cdots &lt;d_{k}=n,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5047f5ab8cc8aed536751a05ec129dc5f9516f5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:33.605ex; height:2.509ex;" alt="0=d_{0}<d_{1}<d_{2}<\\\\cdots <d_{k}=n,"></span></dd></dl>
<br/>where <i>n</i> is the dimension of <i>V</i> (assumed to be finite). Hence, we must have <i>k</i> ≤ <i>n</i>. A flag is called a <b>complete flag</b> if <i>d</i><sub><i>i</i></sub> = <i>i</i> for all <i>i</i>, otherwise it is called a <b>partial flag</b>.
<br/>A partial flag can be obtained from a complete flag by deleting some of the subspaces. Conversely, any partial flag can be completed (in many different ways) by inserting suitable subspaces.
<br/>The <b>signature</b> of the flag is the sequence (<i>d</i><sub>1</sub>, ..., <i>d</i><sub><i>k</i></sub>).
<br/> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Flag_(linear_algebra)">https://en.wikipedia.org/wiki/Flag_(linear_algebra)</a>)"""@en, """En mathématiques, un <b>drapeau</b> d'un espace vectoriel <i>E</i> de dimension finie est une suite finie strictement croissante de sous-espaces vectoriels de <i>E</i>, commençant par l'espace nul {0} et se terminant par l'espace total <i>E</i>&nbsp;: 
<br/>
<br/><center><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 \\\\{0\\\\}=E_{0}\\\\subsetneq E_{1}\\\\subsetneq \\\\cdots \\\\subsetneq E_{k-1}\\\\subsetneq E_{k}=E.}">
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<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\{0\\\\}=E_{0}\\\\subsetneq E_{1}\\\\subsetneq \\\\cdots \\\\subsetneq E_{k-1}\\\\subsetneq E_{k}=E.}</annotation>
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<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1744c84be76c279b8db15800f2c691fd9f60f331" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:40.471ex; height:2.843ex;" alt="{\\\\displaystyle \\\\{0\\\\}=E_{0}\\\\subsetneq E_{1}\\\\subsetneq \\\\cdots \\\\subsetneq E_{k-1}\\\\subsetneq E_{k}=E.}"></span></center>
<br/>Si <i>n</i> est la dimension de <i>E</i>, les dimensions successives des sous-espaces <i>E<sub>i</sub></i> forment une suite finie strictement croissante d'entiers naturels&nbsp;:
<br/>
<br/><center><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 0=d_{0}<d_{1}<\\\\dots <d_{k-1}<d_{k}=n.}">
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<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle 0=d_{0}&lt;d_{1}&lt;\\\\dots &lt;d_{k-1}&lt;d_{k}=n.}</annotation>
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<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac231d44bd65cd38c2df2639706086e1d5407a5e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:35.74ex; height:2.509ex;" alt="{\\\\displaystyle 0=d_{0}<d_{1}<\\\\dots <d_{k-1}<d_{k}=n.}"></span></center>
<br/>Si <i>d<sub>i</sub></i> = <i>i</i> pour tout <i>i</i> (donc entre autres si <i>k</i> = <i>n</i>), alors le drapeau est dit <b>total</b> ou <b>complet</b>. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Drapeau_(math%C3%A9matiques)">https://fr.wikipedia.org/wiki/Drapeau_(math%C3%A9matiques)</a>)"""@fr ;
  skos:prefLabel "drapeau"@fr, "flag"@en .

psr:-M3NJVVTK-V
  skos:prefLabel "homogeneous space"@en, "espace homogène"@fr ;
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