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Area Mathematics - General concepts and linear algebra / Vectors and tensors

IEV ref102-03-16

en
bilinear form
function f that attributes a scalar f(U,V) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaaC yvaiaabYcacaaMe8UaaCOvaiaacMcaaaa@3F0A@ to any pair of vectors U and V in a given vector space, with the following properties:

  • f(αU,V)=αf(U,V) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaeq ySdeMaaGPaVlaahwfacaGGSaGaaGjbVlaahAfacaGGPaGaeyypa0Ja eqySdeMaaGPaVlaadAgacaGGOaGaaCyvaiaacYcacaaMe8UaaCOvai aacMcaaaa@4CA3@ and f(U,βV)=βf(U,V) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaaC yvaiaacYcacaaMe8UaeqOSdiMaaGPaVlaahAfacaGGPaGaeyypa0Ja eqOSdiMaaGPaVlaadAgacaGGOaGaaCyvaiaacYcacaaMe8UaaCOvai aacMcaaaa@4CA7@ where α and β are scalars,
  • f(U+V,W)=f(U,W)+f(V,W) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaaC yvaiabgUcaRiaahAfacaGGSaGaaGjbVlaahEfacaGGPaGaeyypa0Ja amOzaiaacIcacaWHvbGaaiilaiaaysW7caWHxbGaaiykaiabgUcaRi aadAgacaGGOaGaaCOvaiaacYcacaaMe8UaaC4vaiaacMcaaaa@4F34@ and f(W,U+V)=f(W,U)+f(W,V) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaaC 4vaiaacYcacaaMe8UaaCyvaiabgUcaRiaahAfacaGGPaGaeyypa0Ja amOzaiaacIcacaWHxbGaaiilaiaaysW7caWHvbGaaiykaiabgUcaRi aadAgacaGGOaGaaC4vaiaacYcacaaMe8UaaCOvaiaacMcaaaa@4F34@ for any vector W existing in the same vector space

Note 1 to entry: A bilinear form over an n-dimensional vector space can be represented by a square matrix ( k ij ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaabmaabaGaam4Aam aaBaaaleaacaWGPbGaamOAaaqabaaakiaawIcacaGLPaaaaaa@3D59@ and the scalar is f(U,V)= ij k ij U i V j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaaC yvaiaabYcacaaMe8UaaCOvaiaacMcacqGH9aqpdaaeqbqaaiaadUga daWgaaWcbaGaamyAaiaadQgaaeqaaaqaaiaadMgacaWGQbaabeqdcq GHris5aOGaamyvamaaBaaaleaacaWGPbaabeaakiaadAfadaWgaaWc baGaamOAaaqabaaaaa@4AFC@ .

Note 2 to entry: The bilinear forms over a given n-dimensional vector space constitute an n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaamOBa4WaaW baaKqzafqabeaajug4aiaaikdaaaaaaa@3D58@ -dimensional vector space.

Note 3 to entry: The concept of bilinear form extends to "linear form" in the case of one vector and to "multilinear form" (or m-linear form) in the case of an ordered set of m vectors.


fr
forme bilinéaire, f
fonction f qui attribue un scalaire f(U,V) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaaC yvaiaabYcacaaMe8UaaCOvaiaacMcaaaa@3F0A@ à tout couple de vecteurs U et V dans un espace vectoriel donné, avec les propriétés suivantes:

  • f(αU,V)=αf(U,V) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaeq ySdeMaaGPaVlaahwfacaGGSaGaaGjbVlaahAfacaGGPaGaeyypa0Ja eqySdeMaaGPaVlaadAgacaGGOaGaaCyvaiaacYcacaaMe8UaaCOvai aacMcaaaa@4CA3@ et f(U,βV)=βf(U,V) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaaC yvaiaacYcacaaMe8UaeqOSdiMaaGPaVlaahAfacaGGPaGaeyypa0Ja eqOSdiMaaGPaVlaadAgacaGGOaGaaCyvaiaacYcacaaMe8UaaCOvai aacMcaaaa@4CA7@ α et β sont des scalaires,
  • f(U+V,W)=f(U,W)+f(V,W) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaaC yvaiabgUcaRiaahAfacaGGSaGaaGjbVlaahEfacaGGPaGaeyypa0Ja amOzaiaacIcacaWHvbGaaiilaiaaysW7caWHxbGaaiykaiabgUcaRi aadAgacaGGOaGaaCOvaiaacYcacaaMe8UaaC4vaiaacMcaaaa@4F34@ et f(W,U+V)=f(W,U)+f(W,V) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaaC 4vaiaacYcacaaMe8UaaCyvaiabgUcaRiaahAfacaGGPaGaeyypa0Ja amOzaiaacIcacaWHxbGaaiilaiaaysW7caWHvbGaaiykaiabgUcaRi aadAgacaGGOaGaaC4vaiaacYcacaaMe8UaaCOvaiaacMcaaaa@4F34@ pour tout vecteur W du même espace vectoriel

Note 1 à l'article: Une forme bilinéaire sur un espace vectoriel à n dimensions peut être représentée par une matrice carrée ( k ij ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaabmaabaGaam4Aam aaBaaaleaacaWGPbGaamOAaaqabaaakiaawIcacaGLPaaaaaa@3D59@ et le scalaire est f(U,V)= ij k ij U i V j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaaC yvaiaabYcacaaMe8UaaCOvaiaacMcacqGH9aqpdaaeqbqaaiaadUga daWgaaWcbaGaamyAaiaadQgaaeqaaaqaaiaadMgacaWGQbaabeqdcq GHris5aOGaamyvamaaBaaaleaacaWGPbaabeaakiaadAfadaWgaaWc baGaamOAaaqabaaaaa@4AFC@ .

Note 2 à l'article: Les formes bilinéaires sur un espace vectoriel à n dimensions constituent un espace vectoriel à n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaamOBa4WaaW baaKqzafqabeaajug4aiaaikdaaaaaaa@3D58@ dimensions.

Note 3 à l'article: Le concept de forme bilinéaire se généralise à celui de «forme linéaire» dans le cas d'un seul vecteur et de «forme multilinéaire» (ou m-linéaire) dans le cas d'un ensemble ordonné de m vecteurs.


de
Bilinearform, f

es
forma bilineal

ja
双一次形式
双線形形式

pl
forma dwuliniowa
forma biliniowa

pt
forma bilinear

sr
билинеарна форма, ж јд

sv
bilinjär form

zh
双线性型

Publication date: 2008-08
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