mat67-Homework6

# mat67-Homework6 - MAT067 University of California Davis...

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Unformatted text preview: MAT067 University of California, Davis Winter 2007 Homework Set 6: Exercises on Eigenvalues Directions : Please work on all exercises and hand in your solutions to Problems 6 and 7 at the beginning of lecture on February 16, 2006. (Because of the midterm there is only one set of homeworks this week!) As usual, we are using F to denote either R or C , and F [ z ] denotes the set of polynomials with coefficients over F . 1. Let V be a finite-dimensional vector space over F , and let S, T ∈ L ( V ) be linear operators on V with S invertible. Given any polynomial p ( z ) ∈ F [ z ], prove that p ( S ◦ T ◦ S- 1 ) = S ◦ p ( T ) ◦ S- 1 . 2. Let V be a finite-dimensional vector space over C , T ∈ L ( V ) be a linear operator on V , and p ( z ) ∈ C [ z ] be a polynomial. Prove that λ ∈ C is an eigenvalue of the linear operator p ( T ) ∈ L ( V ) if and only if T has an eigenvalue μ ∈ C such that p ( μ ) = λ ....
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