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

IEV ref102-03-37

en
determinant, <of n vectors>
for an ordered set of n vectors in an n-dimensional space with a given base, scalar attributed to this set by the unique multilinear form taking the value 0 when the vectors are linearly dependent and the value 1 for the base vectors

Note 1 to entry: When the coordinates of the n vectors U 1 , U 2 , , U n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfadaWgaaWcba qcLboacaaIXaaaleqaaOGaaeilaiaabccacaWHvbWaaSbaaSqaaKqz GdGaaGOmaaWcbeaakiaabYcacaqGGaGaeSOjGSKaaiilaiaabccaca WHvbWaaSbaaSqaaiaad6gaaeqaaaaa@4634@ are arranged as columns or rows of an n×n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaad6gacqGHxdaTca WGUbaaaa@3CCA@ matrix, the determinant of the vectors is equal to the determinant of the matrix:

det( U 1 , U 2 , , U n ) =| U 11 U 12 U 1n U 21 U 22 U 2n U n1 U n2 U nn | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaacsgacaGGLbGaai iDaiaaysW7caGGOaGaaCyvamaaBaaaleaajug4aiaaigdaaSqabaGc caqGSaGaaeiiaiaahwfadaWgaaWcbaqcLboacaaIYaaaleqaaOGaae ilaiaabccacqWIMaYscaqGSaGaaeiiaiaahwfadaWgaaWcbaGaamOB aaqabaGccaqGPaGaaeiiaiabg2da9maaemaabaqbaeqabqabaaaaae aacaWGvbWaaSbaaSqaaKqzGdGaaGymaiaaigdaaSqabaaakeaacaWG vbWaaSbaaSqaaKqzGdGaaGymaiaaikdaaSqabaaakeaacqWIVlctae aacaWGvbWaaSbaaSqaaKqzGdGaaGymaSGaamOBaaqabaaakeaacaWG vbWaaSbaaSqaaKqzGdGaaGOmaiaaigdaaSqabaaakeaacaWGvbWaaS baaSqaaKqzGdGaaGOmaiaaikdaaSqabaaakeaacqWIVlctaeaacaWG vbWaaSbaaSqaaKqzGdGaaGOmaSGaamOBaaqabaaakeaacqWIUlstae aacqWIUlstaeaacqWIXlYtaeaacqWIUlstaeaacaWGvbWaaSbaaSqa aiaad6gajug4aiaaigdaaSqabaaakeaacaWGvbWaaSbaaSqaaiaad6 gajug4aiaaikdaaSqabaaakeaacqWIVlctaeaacaWGvbWaaSbaaSqa aiaad6gacaWGUbaabeaaaaaakiaawEa7caGLiWoaaaa@8131@

Note 2 to entry: According to the sign of the determinant, the set of vectors and the given base have the same orientation or opposite orientations.

Note 3 to entry: For the three-dimensional Euclidean space, the determinant of three vectors is the scalar triple product of the vectors.


fr
déterminant, <de n vecteurs> m
pour un ensemble ordonné de n vecteurs dans un espace à n dimensions muni d'une base donnée, scalaire attribué à cet ensemble par la seule forme multilinéaire qui prend la valeur 0 lorsque les vecteurs sont linéairement dépendants et la valeur 1 pour les vecteurs de base

Note 1 à l'article: Lorsque les coordonnées des n vecteurs U 1 , U 2 , , U n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfadaWgaaWcba qcLboacaaIXaaaleqaaOGaaeilaiaabccacaWHvbWaaSbaaSqaaKqz GdGaaGOmaaWcbeaakiaabYcacaqGGaGaeSOjGSKaaiilaiaabccaca WHvbWaaSbaaSqaaiaad6gaaeqaaaaa@4634@ sont disposés selon les colonnes ou les lignes d'une matrice n×n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaad6gacqGHxdaTca WGUbaaaa@3CCA@ , le déterminant des vecteurs est égal au déterminant de la matrice:

det( U 1 , U 2 , , U n ) =| U 11 U 12 U 1n U 21 U 22 U 2n U n1 U n2 U nn | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaacsgacaGGLbGaai iDaiaaysW7caGGOaGaaCyvamaaBaaaleaajug4aiaaigdaaSqabaGc caqGSaGaaeiiaiaahwfadaWgaaWcbaqcLboacaaIYaaaleqaaOGaae ilaiaabccacqWIMaYscaqGSaGaaeiiaiaahwfadaWgaaWcbaGaamOB aaqabaGccaqGPaGaaeiiaiabg2da9maaemaabaqbaeqabqabaaaaae aacaWGvbWaaSbaaSqaaKqzGdGaaGymaiaaigdaaSqabaaakeaacaWG vbWaaSbaaSqaaKqzGdGaaGymaiaaikdaaSqabaaakeaacqWIVlctae aacaWGvbWaaSbaaSqaaKqzGdGaaGymaSGaamOBaaqabaaakeaacaWG vbWaaSbaaSqaaKqzGdGaaGOmaiaaigdaaSqabaaakeaacaWGvbWaaS baaSqaaKqzGdGaaGOmaiaaikdaaSqabaaakeaacqWIVlctaeaacaWG vbWaaSbaaSqaaKqzGdGaaGOmaSGaamOBaaqabaaakeaacqWIUlstae aacqWIUlstaeaacqWIXlYtaeaacqWIUlstaeaacaWGvbWaaSbaaSqa aiaad6gajug4aiaaigdaaSqabaaakeaacaWGvbWaaSbaaSqaaiaad6 gajug4aiaaikdaaSqabaaakeaacqWIVlctaeaacaWGvbWaaSbaaSqa aiaad6gacaWGUbaabeaaaaaakiaawEa7caGLiWoaaaa@8131@

Note 2 à l'article: Selon le signe du déterminant, l'ensemble de vecteurs et la base donnée ont la même orientation ou des orientations contraires.

Note 3 à l'article: Pour l'espace euclidien à trois dimensions, le déterminant de trois vecteurs est le produit mixte des vecteurs.


de
Determinante (von n Vektoren), f

es
determinante (de n vectores)

ja
nベクトルの行列式

pl
wyznacznik (n wektorów)

pt
determinante (de n vectores)

sr
детерминанта, <n вектора> ж јд

sv
determinant (av n vektorer)

zh
行列式, <n个向量的>

Publication date: 2008-08
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