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College Algebra Exam Review 172

College Algebra Exam Review 172 - 182 3 PRODUCTS OF GROUPS...

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Unformatted text preview: 182 3. PRODUCTS OF GROUPS Definition 3.4.14. We say that two linear transformations T;T of V are similar if there exists an invertible linear transformation S such that T D ST S 1 . We say that two n –by– n matrices A;A are similar if there exists an invertible n –by– n matrix Q such that A D QA Q 1 . Proposition 3.4.15. (a) Let V be a finite dimensional vector space over a field K . The matrices of a linear transformation T 2 End K .V / with respect to two different ordered bases are similar. (b) Conversely, if A and A are similar matrices, then there exists a linear transformation T of a vector space V and two ordered bases B;B of V such that A D OETŁ B and A D OETŁ B . Proof. Part (a) was proved above. For part (b), let A be an n –by– n matrix, Q an invertible n –by– n ma- trix, and set A D QAQ 1 . Let E D . O e 1 ;:::; O e n / be the standard ordered basis of K n , and let B D Q 1 O e 1 ;:::;Q 1 O e n ; thus B consists of the columns of Q 1 ....
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