s11_mthsc853_hw06 - MthSc 853 (Linear Algebra) Dr. Macauley...

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Unformatted text preview: MthSc 853 (Linear Algebra) Dr. Macauley HW 6 Due Wednesday, March 2nd, 2011 (1) Let S n denote the set of all permutations of { 1 ,...,n } . (a) Prove that if S n is a transposition, then sgn( ) =- 1. (b) Let S n , and suppose that = k 1 = 1 , where i , j S n are transpositions. Prove that k = mod 2. (2) Let X be an n-dimensional vector space over a field K . (a) Prove that if the characteristic of K is not 2, then every skew-symmetric form is alternating. (b) Give an example of a non-alternating skew-symmetric form. (c) Give an example of a non-zero alternating k-linear form ( k < n ) such that f ( x 1 ,...,x k ) = 0 for some set of linearly independent vectors x 1 ,...,x k . (3) Let X be a 2-dimensional vector space over C , and let f : X X C be an alternating, bilinear form. If { x 1 ,x 2 } is a basis of X , determine a formula for f ( u,v ) in terms of f ( x 1 ,x 2 ), and the coefficients used to express...
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