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College Algebra Exam Review 171

College Algebra Exam Review 171 - 3.4 THE DUAL OF A VECTOR...

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3.4. THE DUAL OF A VECTOR SPACE AND MATRICES 181 Thus the j th column of OEST Ł D;B and of OESŁ D;C OET Ł C;B agree. n For a vector space V over a field K , we denote the set of K –linear maps from K to K by End K .V / . Since the composition of linear maps is linear, End K .V / has a product .S; T / 7! ST . The reader can check that End K .V / with the operations of addition and and composition of linear operators is a ring with identity. To simplify notation, we write OET Ł B in- stead of OET Ł B;B for the matrix of a linear transformation T with respect to a single basis B of V . Corollary 3.4.12. Let V be a finite dimensional vector space over K . Let n denote the dimension of V and let B be an ordered basis of V . (a) For all S; T 2 End K .V / , OEST Ł B;B D OESŁ B OET Ł B . (b) T 7! OET Ł B is a ring isomorphism from End K .V / to Mat n .K/ . Lemma 3.4.13. Let B D .v 1 ; : : : ; v n / and C D .w 1 ; : : : ; w n / be two bases of a vector space V over a field K . Denote the dual bases of V by B D v 1 ; : : : ; v n and C D w 1 ; : : : ; w n . Let id denote the identity linear transformation of
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