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 τ ∈ S n is a transposition, then sgn( τ ) = 1. (b) Let π ∈ S n , and suppose that π = τ k ◦ ··· ◦ τ 1 = σ ‘ ◦ ··· ◦ σ 1 , where τ i ,σ j ∈ S n are transpositions. Prove that k ∼ = ‘ mod 2. (2) Let X be an ndimensional vector space over a field K . (a) Prove that if the characteristic of K is not 2, then every skewsymmetric form is alternating. (b) Give an example of a nonalternating skewsymmetric form. (c) Give an example of a nonzero alternating klinear 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 2dimensional 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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 Spring '08
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 Linear Algebra, Algebra, Permutations, Vector Space, ndimensional vector space, Dr. Macauley HW

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