mat67-Homework6_Solutions

mat67-Homework6_Solutions - MAT067 University of...

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Unformatted text preview: MAT067 University of California, Davis Winter 2007 Solutions to Homework Set 6 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 . Solution : Proposition. Let V be a finite-dimensional vector space over F with S, T ∈ L ( V ) linear operators on V such that S invertible. Given p ( z ) ∈ F [ z ] a polynomial, p ( S ◦ T ◦ S- 1 ) = S ◦ p ( T ) ◦ S- 1 . Proof. Let p ( z ) ∈ F [ z ] be a polynomial with coefficients in F , and suppose the p ( z ) = ∑ n k =0 a k z k where a k ∈ F and n = deg( p ). Then, note that for each monomial a k z k in p , the corresponding monomial in p ( S ◦ T ◦ S- 1 ) is a k ( S ◦ T ◦ S- 1 ) k = a k ( S ◦ T ◦ S- 1 ) · ··· · ( S ◦ T ◦ S- 1 ) bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright k times = ··· = a k S ◦ T k ◦ S- 1 , where we have repeatedly used the fact that S- 1 ◦ S is the identity operator on V . Thus, it follows that p ( S ◦ T ◦ S- 1 ) can equivalently be written as n summationdisplay k =0 a k ( S ◦ T ◦ S- 1 ) k = n summationdisplay k =0 a k S ◦ T k ◦ S- 1 = S ◦ parenleftBigg n summationdisplay k =0 a k T k parenrightBigg ◦ S- 1 = S ◦ p ( T ) ◦ S- 1 . 2 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 . Solution : Proposition. Let V be a finite-dimensional vector space over C and T ∈ L ( V ) be a linear operator on V . Then, given any polynomial p ( z ) ∈ C [ z ] , 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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This homework help was uploaded on 04/08/2008 for the course MATH 67 taught by Professor Schilling during the Fall '07 term at UC Davis.

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mat67-Homework6_Solutions - MAT067 University of...

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