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Unformatted text preview: 9) Given a linear operator T has matrix representaiton A = 6- 3- 2 4- 1- 2 10- 5- 3 , express its minimal polynomial as the product of monic irreducible polynomials, p 1 p 2 . Simple computations confirm that the minimal polynomial for T is ( X- 2 I )( X 2 + I ), so p 1 ( X ) = X- 2 I , p 2 ( X ) = X 2 + I . 9) Given a linear operator T has matrix representaiton A = 3 1- 1 2 2- 1 2 2 , show that there is a diagonalizable operator D and a nilpotent operator N on R 3 such that T = D + N and ND = DN . Find the matrices for D and N . 9) Let V be the vectorspace of all polynomials of degree n over a field F . Show that the differentiation operator is nilpotent. 9) Let T be a linear operator on a finite dimensional vectorspace V with minimal polynomial f = ( x- c 1 ) d 1 ( x- c k ) d k . Let W i be the null space of ( T- c i I ) d i . (a) Show that the set S of vectors b such that there exists some m for which ( T- c i I ) m b = 0 is equal to the subspace...
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This document was uploaded on 01/25/2012.
- Spring '09