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 KŒxŁ module, ˚ W F ±! V is a surjective KŒxŁ –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 ² j ² s , and V j D KŒxŁv j D span . f p.T /v j W p.x/ 2 KŒxŁ g /: (a) Show that V D V 1 ˚ ³³³ ˚ 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 ² 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 .A/ , namely
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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 is a root of T .x/ if, and only if, T has an eigenvector in V with eigenvalue . 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 eld K contains all roots of T .x/ ; that is, T .x/ factors into linear factors in Kx . 8.6.8. Let V V be any invariant subspace for T . Show that there is a linear operator T on V=V dened 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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