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College Algebra Exam Review 395

# College Algebra Exam Review 395 - the sum of the matrix...

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8.6. RATIONAL CANONICAL FORM 405 8.6.3. Verify the following assetions made in the text regarding the com- putation of the rational canonical form of T . Suppose that F is a free KOExŁ module, ˚ W F ! V is a surjective KOExŁ –module homomorphism, .y 1 ; : : : ; y n s ; z 1 ; : : : ; z s / is a basis of F , and .y 1 ; : : : ; y n s ; a 1 .x/z 1 ; : : : ; a s .x/z s / is a basis of ker .˚/ . Set v j D ˚.z j / for 1 j s , and V j D KOExŁv j D span . f p.T /v j W p.x/ 2 KOExŁ g /: (a) Show that V D V 1 ˚ ˚ V s . (b) Let ı j be the degree of a j .x/ . Show that v j ; T v j ; : : : ; T ı j 1 v j is a basis of V j . and that the matrix of T j V j with respect to this basis is the companion matrix of a j .x/ . 8.6.4. Let A D 2 6 6 4 7 4 5 1 15 10 15 3 0 0 5 0 56 52 51 15 3 7 7 5 . Find the rational canonical form of A and find an invertible matrix S such that S 1 AS is in rational canonical form. 8.6.5. Show that A is a similarity invariant of matrices. Conclude that for T 2 End K .V / , T is well defined, and is a similarity invariant for linear transformations. 8.6.6. Since A .x/ is a similarity invariant, so are all of its coefficients. Show that the coefficient of x n 1 is the negative of the trace tr
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Unformatted text preview: the sum of the matrix entries on the main diagonal of A . Conclude that the trace is a similarity invariant. 8.6.7. Show that ² is a root of ± T .x/ if, and only if, T has an eigenvector in V with eigenvalue ² . Show that v is an eigenvector of T for some eigen-value if, and only if, the one dimensional subspace Kv ´ V is invariant under T . The next four exercises give an alternative proof of the Cayley-Hamilton theorem. Let T 2 End K .V / , where V is n –dimensional. Assume that the ﬁeld K contains all roots of ± T .x/ ; that is, ± T .x/ factors into linear factors in KŒxŁ . 8.6.8. Let V ´ V be any invariant subspace for T . Show that there is a linear operator T on V=V deﬁned by T .v C V / D T.v/ C V for all v 2 V . Suppose that .v 1 ;:::;v k / is an ordered basis of V , and that .v k C 1 C V ;:::;v n C V /...
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