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Unformatted text preview: 7 Let T be the linear operator on R 4 which is represented in the standard ordered basis by the matrix M = 0 0 0 a 0 0 0 b 0 0 c Under what conditions on a,b,c is T diagonalizable? 7 Let T be a linear operator on the n-dimensional vector space V , and suppose that T has n distinct characteristic values. Prove that T is diagonalizable. 7 Let A and B F n n . Prove that if ( I- AB ) is invertible then ( I- BA ) is invertible with inverse I + B ( I- AB )- 1 A . 7 Prove for A,B F n n that AB and BA have precisely the same characteristic values in F . 7 Suppose that A R 2 2 is symmetric. Prove that A is similar over R to a diagonal matrix. 7 Let N C 2 2 such that N 2 = 0. Prove that either N = 0 or N is similar over C to 0 0 1 0 . 7 If A C 2 2 show that A is similar over C to either a diagonal matrix or a matrix of the form s 1 s ....
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This document was uploaded on 01/25/2012.
- Spring '09
- Vector Space