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IEVref:102-03-08ID:
Language:enStatus: Standard
Term: base, <in linear algebra>
Synonym1: basis
[Preferred]
Synonym2:
Synonym3:
Symbol:
Definition: ordered set of n linearly independent vectors a 1 , a 2 ,..., a n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahggadaWgaaWcba acbaqcLboacaWFXaaaleqaaOGaaeilaiaaysW7caWHHbWaaSbaaSqa aKqzGdGaa8NmaaWcbeaakiaabYcacaaMe8UaaeOlaiaab6cacaqGUa GaaeilaiaaysW7caWHHbWaaSbaaSqaaiaad6gaaeqaaaaa@49FD@ in an n-dimensional vector space, which is chosen to express any vector U as a unique linear combination of these n vectors

U= U 1 a 1 + U 2 a 2 +...+ U n a n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacqGH9aqpca WGvbWaaSbaaSqaaGqaaKqzGdGaa8xmaaWcbeaakiaahggadaWgaaWc baqcLboacaWFXaaaleqaaOGaey4kaSIaamyvamaaBaaaleaajug4ai aa=jdaaSqabaGccaWHHbWaaSbaaSqaaKqzGdGaa8NmaaWcbeaakiab gUcaRiaac6cacaGGUaGaaiOlaiabgUcaRiaadwfadaWgaaWcbaGaam OBaaqabaGccaWHHbWaaSbaaSqaaiaad6gaaeqaaaaa@5010@ , where U 1 , U 2 ,..., U n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadwfadaWgaaWcba acbaqcLboacaWFXaaaleqaaOGaaeilaiaaysW7caWGvbWaaSbaaSqa aKqzGdGaa8NmaaWcbeaakiaabYcacaaMe8UaaeOlaiaab6cacaqGUa GaaeilaiaaysW7caWGvbWaaSbaaSqaaiaad6gaaeqaaaaa@49CD@ are scalars

Note 1 to entry: In an Euclidean or Hermitian vector space, an orthonormal base is generally chosen. In the vector space formed by a set of n-bit words (see Note 1 to entry in IEV 102-03-01, vector space) a base is the set of n-bit words having only one non-zero bit.

Note 2 to entry: Any vector of a base is called "base vector".


Publication date:2017-07
Source:
Replaces:102-03-08:2007-08
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