# quiz5 - Quiz #5 1. Label the following statements as being...

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Quiz #5 1. Label the following statements as being true or false. Assume that V is a ﬁnite-dimensional inner product space (a) An inner product is linear in both components. (Solution) False. h x,cy i = ¯ c h x,y i . (b) If x , y , and z are vectors in V such that h x,y i = h x,z i , then y = z . (Solution) False. Take x = 0. (c) If h x,y i = 0 for all x in V , then y = 0. (Solution) True. Take x = y . (d) Every ﬁnite-dimensional inner product space has an orthonormal basis. (Solution) True. Use Gram-Schmidt orthogonalization process to get an orthogonal basis from the given basis. And then nor- malize it! (e) The orthogonal complement of any set is a subspace. (Solution) True. easy to prove it. Try! (f) Every orthogonal set is linearly independent. (Solution) False. Consider the zero vector (g) Every orthonormal set is linearly independent. (Solution) True. Try to prove it! (h) For every linear operator T on V and every basis β for V , we have [ T * ] β = ([ T ] β ) * .

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## quiz5 - Quiz #5 1. Label the following statements as being...

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