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Unformatted text preview: K of Mat n ( k ) , consisting of all the skew symmetric matrices, is a subspace of Mat n ( k ) . Solution. Now 0 is skew-symmetric, for 0 T = = 0. If A and B are skew-symmetric, then ( A + B ) T = A T + B T = A B = ( A + B ), and if is a scalar, then ( A ) T = ( A T ) = A . Therefore, K is a subspace of Mat n ( k ) . (ii) Determine dim ( K ) . Solution. If A is skew symmetric, then all its diagonal entries are 0. The answer is 1 2 ( n 2 n ) . 4.13 If p is a prime with p 1 mod 4, prove that there is a nonzero vector v F 2 p with (v, v) = 0, where (v, v) is the usual inner product of v with itself [see Example 4.4(i)]. Solution. Use the Two-squares theorem....
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- Fall '11