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Unformatted text preview: 396 8. MODULES where V i is a cyclic KOEx module V i KOEx=.a i .x//; and the polynomials a i .x/ are the invariant factors of T . By Lemma 8.6.2 , there is a basis of V i such that the matrix of T i with respect to this basis is the companion matrix of a i .x/ . Therefore, there is a basis of V with respect to which the matrix of T is in rational canonical form. Now suppose that the matrix A of T with respect to some basis is in rational canonical form, with blocks C a i for 1 i s . It follows that .T;V / has a direct sum decomposition .T;V / D .T 1 ;V 1 / .T s ;V s /; where the matrix of T i with respect to some basis of V i is C a i . By Lemma 8.6.2 , V i KOEx=.a i .x// as KOEx modules. Thus V KOEx=.a 1 .x// KOEx=.a s .x//: By the uniqueness of the invariant factor decomposition of V (Theorem 8.5.2 ), the polynomials a i .x/ are the invariant factors of the KOEx module V , that is, the invariant factors of T . Thus the polynomials a i .x/ , and...
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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.
 Fall '08
 EVERAGE
 Algebra, Polynomials, Factors

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