hw09 - Math 110—Linear Algebra Fall 2009 Haiman Problem...

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Unformatted text preview: Math 110—Linear Algebra Fall 2009, Haiman Problem Set 9 Due Monday, Nov. 2 at the beginning of lecture. 1. Prove that if A and Q are n × n matrices over F , with Q invertible, then det( Q- 1 AQ ) = det( A ). Deduce that if V is a finite-dimensional vector space and T : V → V is a linear transformation, then det([ T ] β ) does not depend on the choice of the ordered basis β of V . 2. A matrix of the form A = 1 x 1 x 2 1 ... x n- 1 1 1 x 2 x 2 2 ... x n- 1 2 . . . . . . . . . . . . 1 x n x 2 n ... x n- 1 n is called a Vandermonde matrix. (a) Show that the determinant det( A ) is a polynomial in the variables x 1 ,x 2 ,...,x n in which every term has degree n ( n- 1) / 2. (The degree of a monomial x a 1 1 x a 2 2 ··· x a n n is defined to be a 1 + ··· + a n .) (b) Show that det( A ) becomes zero if x i = x j for any i and j . This implies that det( A ) is divisible as a polynomial in the x i ’s by the product Y 1 ≤ i<j ≤ n ( x j- x...
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hw09 - Math 110—Linear Algebra Fall 2009 Haiman Problem...

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