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College Algebra Exam Review 389

College Algebra Exam Review 389 - 8.6 RATIONAL CANONICAL...

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8.6. RATIONAL CANONICAL FORM 399 with basis f e 1 ; : : : ; e n g . Let A D .a i;j / be the matrix of T with respect to this basis, so Te j D P i a i;j e i . Let F be the free KOExŁ –module with basis f f 1 ; : : : ; f n g and define ˚ W F ! V by P j h i .x/f i 7! P j h i .T /e i . Then ˚ is a surjective KOExŁ –module homomorphism. We need to find the kernel of ˚ . The transformation T can be “lifted” to a KOExŁ –module homomor- phism of F by using the matrix A . Define T W F ! F by requiring that Tf j D P i a i;j f i . Then we have ˚.Tf / D T ˚.f / for all f 2 F . I claim that the kernel of ˚ is the range of x T . This follows from three observations: 1. range .x T / ker .˚/ . 2. range .x T / C F 0 D F , where F 0 denotes the set of K –linear combinations of f f 1 ; : : : ; f n g . 3. ker .˚/ \ F 0 D f 0 g . The first of these statements is clear since ˚.xf / D ˚.Tf / D T ˚.f / for all f 2 F . For the second statement, note that for any h.x/ 2 KOExŁ , h.x/f j D .h.x/ h.T //f j C h.T /f j : Since multiplication by x and application of T commute, there is a poly-
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