Problem 1. Indicate whether the statement is true(T) or (5) If A is a symmetric matrix, then eigenvectors from dierent eigenspaces are orthogonal. (T) false(F). Justify your answer. [each 3pt] (1) If T : Rn ¡æ Rn is a linear operator, and if [T ]B = [T ]B with respect to two bases B and B for Rn , then B = B . (F) Solve If T is a zero operator, then [T ]B = O for any basis for R . So [T ]B = [T ]B but B = B . So (¥ë1 ¥ë2 )(x1 x2 ) = 0 and thus x1 x2 = 0.
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solve Suppose that x1 ¡ô E¥ë1 and x2 ¡ô E¥ë2 are eigenvectors from dierent eigenspaces. Then, (¥ë1 x1 ) x2 = (Ax1 ) x2 = x1 (AT x2 )
= x1 (Ax2 ) = x1 (¥ë2 x2 )
(2) If V and W are distinct subspaces of Rn with the same dimension, then neither V nor W is a subspace of the other. (T) Solve With out loss of generality, if V is a subspace of W . Since dim(V ) = dim(W ) and V is a subspace of W , V = W . It is a contradiction. Problem 2. Indicate whether the statement is true(T) or false(F). [each 2pt] (1) If A = U ¥ÒV ¡¦(»ý·«)
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