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College Algebra Exam Review 173

# College Algebra Exam Review 173 - Exercises 3.4 3.4.1...

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3.4. THE DUAL OF A VECTOR SPACE AND MATRICES 183 to some basis of V . Since the matrices of T with respect to two different bases are similar, the result does not depend on the choice of the basis. Two important similarity invariants are the determinant and the trace. Because the determinant satisfies det .AB/ D det .A/ det .B/ , and det .E/ D 1 , it follows that det .C 1 / D det .C/ 1 and det .CAC 1 / D det .C/ det .A/ det .C/ 1 D det .A/: Thus determinant is a similarity invariant. 4 The trace of a square matrix is the sum of its diagonal entries. Let A D .a i;j / . Let C D .c i;j / be an invertible matrix and let C 1 D .d i;j / . Since .d i;j / and .c i;j / are inverse matrices, we have P i d k;i c i;j D ı kj for any k; j . Using this, we compute: tr .CAC 1 / D X i .CAC 1 /.i; i/ D X i X j X k c i;j a j;k d k;i D X j X k . X i d k;i c i;j /a j;k D X j X k ı k;j a j;k D X j a j;j D tr .A/: Thus the trace is also a similarity invariant.
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Unformatted text preview: Exercises 3.4 3.4.1. Complete the details of the veriﬁcation that Hom K .V;W / is a K – vector space, when V and W are K –vector spaces. 3.4.2. Consider the ordered basis B D @ 2 4 1 1 1 3 5 ; 2 4 1 1 3 5 ; 2 4 1 3 5 1 A of R 3 . Find the dual basis of . R 3 / ² . 3.4.3. Deﬁne a bilinear map from K n ± K n to K by Œ 2 6 6 6 4 ˛ 1 ˛ 2 : : : ˛ n 3 7 7 7 5 ; 2 6 6 6 4 ˇ 1 ˇ 2 : : : ˇ n 3 7 7 7 5 Ł D n X j D 1 ˛ j ˇ j : Show that the induced map ± W K n ²! .K n / ² given by ±. v /. w / D Œ w ; v Ł is an isomorphism. 4 The determinant is dicussed systematically in Section 8.3 ....
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