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IEVref:102-03-37ID:
Language:enStatus: Standard
Term: determinant, <of <i>n</i> vectors>
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Definition: 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 nn 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.


Publication date:2008-08
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Internal notes:2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO
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