College Algebra Exam Review 386

College Algebra Exam Review 386 - 396 8. MODULES where V i...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

Ask a homework question - tutors are online