College Algebra Exam Review 385

College Algebra Exam Review 385 - if it is block diagonal 2...

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8.6. RATIONAL CANONICAL FORM 395 Write J D .a.x// for convenience. I claim that B D ± 1 C J;x C J;:::;x d ± 1 C J ² is a basis of KŒxŁ=.a.x// over K . In fact, for any f.x/ 2 KŒxŁ , we can write f.x/ D q.x/a.x/ C r.x/ where r.x/ D 0 or deg .r.x// < d . Then f.x/ C J D r.x/ C J , which means that B spans KŒxŁ=.a.x// over K . If B is not linearly independent, then there exists a nonzero polynomial r.x/ of degree less than d such that r.x/ 2 J ; but this is impossible since J D .a.x// . The matrix of T with respect to B is clearly the companion matrix of a.x/ , as T.x j C J/ D x j C 1 C J for j ± d ² 2 and T.x d ± 1 C J/ D x d C J D ² .a 0 C a 1 x C ³³³ C a d ± 1 x d ± 1 / C J . Finally, if V has a basis B D .v 0 ;:::;v d ± 1 / with respect to which the matrix of T is the companion matrix of a.x/ , then v j D T j v 0 for j ± d ² 1 and T d v 0 D T v d ± 1 D ² . P d ± 1 i D 0 ˛ i v i / D ² . P d ± 1 i D 0 ˛ i T i /v 0 . Therefore, V is cyclic with generator v 0 and a.x/ 2 ann .v 0 / . No polyno- mial of degree less than d annihilates v 0 , since f T j v 0 W j ± d ² 1 g D B is linearly independent. This shows that condition (d) implies (b). n Definition 8.6.3. Say that a matrix is in rational canonical form
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Unformatted text preview: if it is block diagonal 2 6 6 6 4 C a 1 ³³³ C a 2 ³³³ : : : : : : : : : ³³³ C a s 3 7 7 7 5 ; where C a i is the companion matrix of a monic polynomial a i .x/ of degree ´ 1 , and a i .x/ divides a j .x/ for i ± j Theorem 8.6.4. (Rational canonical form) Let T be a linear transforma-tion of a finite dimensional vector space V over a field K . (a) There is an ordered basis of V with respect to which the matrix of T is in rational canonical form. (b) Only one matrix in rational canonical form appears as the matrix of T with respect to some ordered basis of V . Proof. According to Theorem 8.5.16 , .T;V / has a direct sum decomposi-tion .T;V / D .T 1 ;V 1 / ˚ ³³³ ˚ .T s ;V s /;...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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