College Algebra Exam Review 165

College Algebra Exam Review 165 - 3.4 THE DUAL OF A VECTOR...

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3.4. THE DUAL OF A VECTOR SPACE AND MATRICES 175 Equivalently, the coordinate vector of f with respect to the orderd basis B ± is S B ± .f / D 2 6 6 6 4 f.v 1 / f.v 2 / : : : f.v n / 3 7 7 7 5 For v 2 V , the expansion of v in terms of the basis B is expressed with the help of the dual basis B ± as v D n X j D 1 v ± j .v/v j : Equivalently, the coordinate vector of v with respect to the ordered basis B is S B .v/ D 2 6 6 6 4 v ± 1 .v/ v ± 2 .v/ : : : v ± n .v/ 3 7 7 7 5 In fact, this is clear because for v D P n j D 1 ˛ j v j , we have ˛ j D v ± j .v/ for each j , and therefore v D P n j D 1 v ± j .v/v j . We have proved: Proposition 3.4.2. Let V be a finite dimensional vector space with basis B D f v 1 ;v 2 ;:::;v n g . (a) For each j ( 1 ± j ± n ), there is a linear functional v ± j on V determined by v ± j .v i / D ı
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