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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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 Spring '09
 Linear Algebra, Polynomials, Vector Space, 1 k, WI

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