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College Algebra Exam Review 387

College Algebra Exam Review 387 - 8.6 RATIONAL CANONICAL...

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8.6. RATIONAL CANONICAL FORM 397 and only if, the KOExŁ –modules determined by these linear transformations are isomorphic as KOExŁ –modules. Let V 1 denote V endowed with the KOExŁ –module structure derived from T 1 and let V 2 denote V endowed with the KOExŁ –module structure derived from T 2 . Suppose U W V 1 ! V 2 is a KOExŁ –module isomorphism; then U is a vector space isomorphism satisfying T 2 .Uv/ D x.Uv/ D U.xv/ D U.T 1 v/ . It follows that T 2 D U T 1 U 1 . Conversely, suppose that U is an invertible linear transformation such that T 2 D U T 1 U 1 . It follows that for all f .x/ 2 KOExŁ , f .T 2 / D Uf .T 1 /U 1 ; equivalently, f .T 2 /Uv D Uf .T 1 /v for all v 2 V But this means that U is a KOExŁ –module isomorphism from V 1 to V 2 . n Rational canonical form for matrices Let A be an n –by– n matrix over a field K . Let T be the linear trans- formation of K n determined by left multiplication by A , T .v/ D Av for v 2 K n . Thus, A is the matrix of T with respect to the standard basis of K n . A second matrix A 0 is similar to A if, and only if, A 0 is the matrix of T with respect to some other ordered basis. Exactly one such matrix is in rational canonical form, according to Theorem
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