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 ŒT 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 ŒT 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 ŒT Ł C;B is the following: ŒT Ł C;B is the standard matrix of S C TS ² 1 B W K m ±! K n : Here we are indicating com- position of linear maps by juxtaposition; i.e., S C TS ² 1 B D S C ı T ı S ² 1 B . As discussed in Appendix E , the standard matrix M of a linear transformation
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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