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**Unformatted text preview: **} is a subspace of F n . (b) Now suppose that m = n and A is invertible, and that B = { ~v 1 ,...,~v k } is a basis for W . Show that { A-1 ~v 1 ,...,A-1 ~v k } is a basis for U . 4. An n × n matrix A (over the ﬁeld F ) is called symmetric if and only if A = A T . A is called antisymmetic (or skew-symmetric ) just if A T =-A . Fix n ≥ 2 and let U be the set of all n × n symmetric matrices over F , and W be the set of all n × n antisymmetric matrices over F . (a) Show that each of U and W is a subspace of M n ( F ). For the rest of the problem, V = M n ( F ), and F is either R or C . (b) Show that V = U ⊕ W . What happens if F = Z 2 ? (c) In case n = 3, ﬁnd a basis for each of U and W . (d) Find the dimension of each of U and W for any n . 1...

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