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College Algebra Exam Review 169

College Algebra Exam Review 169 - 3.4 THE DUAL OF A VECTOR...

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3.4. THE DUAL OF A VECTOR SPACE AND MATRICES 179 Corollary 3.4.9. . dim W C dim W ı D dim V . Proof. Exercise 3.4.9 . n Matrices Let V and W be finite dimensional vector spaces over a field K . Let B D .v 1 ; : : : ; v m / be an ordered basis of V and C D .w 1 ; : : : ; w n / an ordered basis of W . Let C D .w 1 ; : : : ; w n / denote the basis of W dual to C . Let T 2 Hom K .V; W / . The matrix OET Ł C;B of T with respect to the ordered bases B and C is the n –by– m matrix whose .i; j / entry is h T v j ; w i i . Equivalently, the j –th column of the matrix OET Ł C;B is S C .T .v j // D 2 6 6 6 4 h T .v j /; w 1 i h T .v j /; w 2 i : : : h T .v j /; w n i 3 7 7 7 5 ; the coordinate vector of T .v j / with respect to the ordered basis C . Another useful description of OET Ł C;B is the following: OET Ł C;B is the standard matrix of S C TS 1 B W K m ! K n : Here we are indicating com- position of linear maps by juxtaposition; i.e., S C TS 1 B D S C ı T ı S 1 B . As discussed in Appendix E , the standard matrix M of a linear transformation
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