# Adv Alegbra HW Solutions 135 - A is symmetric Solution...

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135 (ii) Prove that a linear transformation U : R n R n is orthogonal if and only if U (v 1 ),. .., U (v n ) is an orthonormal basis whenever v 1 ,...,v n is an orthonormal basis. Solution. Absent. (iii) If w R n and v 1 ,...,v n is an orthonormal basis, then w = n i = 1 c i v i . Prove that c i = (w, v i ) . Solution. Absent. 4.44 Let U : R n R n be an orthogonal transformation, and let X = v 1 ,...,v n be an orthonormal basis. If O = X [ U ] X , prove that O 1 = O T . Solution. Absent. 4.45 Let A be an n × n real symmetric matrix. (i) Give an example of a nonsingular matrix P for which PAP 1 is not symmetric. Solution. Absent. (ii) Prove that OAO 1 is symmetric for every n × n real orthogonal matrix O . Solution. Absent. 4.46 True or false with reasons. (i) If a matrix A is similar to a symmetric matrix, then
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Unformatted text preview: A is symmetric. Solution. False. (ii) £ 2 1 0 1 ± is invertible over Q . Solution. True. (iii) £ 2 1 0 1 ± is invertible over Z . Solution. False. (iv) If A is a 2 × 2 matrix over R all of whose entries are positive, then det ( A ) is positive. Solution. False. (v) If A is a 2 × 2 matrix over R all of whose entries are positive, then det ( A ) ≥ 0. Solution. False. (vi) If A and B are n × n matrices, then tr ( A + B ) = tr ( A ) + tr ( B ) . Solution. True. (vii) If two n × n matrices over a f eld k have the same characteristic polynomial, then they are similar. Solution. False....
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