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**Unformatted text preview: **Math 21b Midterm IISolutions Thursday, April 10, 2003 1. (12 points) True or False. No justification is necessary, simply circle T or F for each statement. T F (a) If A is an n n invertible matrix, then det( A T A ) > 0. Solution. True. det( A T A ) = det( A T ) det( A ) = [det( A )] 2 > 0. T F (b) The subset W = { f C ( R ) : f 00 ( x ) + f ( x ) = x 2 } is a subspace of the linear space of all infinitely differentiable functions C ( R ). Solution. False. f 0 is not in W . T F (c) If v R n and W is a subspace of R n , then k v k 2 = k proj W v k 2 + k v- proj W v k 2 . Solution. True. proj W v v- proj W v . T F (d) If det( A ) 6 = det( B ), then two n n matrices A and B cannot be similar. Solution. True. Consider the contrapositive. If A and B are similar matrices, then det( B ) = det( S- 1 AS ) = det( S- 1 ) det( A ) det( S ) = det( A ). T F (e) The vectors x +1, x- 1, and x 2- 1 are linearly dependent in P 2 , the polynomials of degree less than or equal to 2. Solution. False. Let 0 = c 1 ( x + 1) + c 2 ( x- 1) + c 3 ( x 2- 1) = ( c 1- c 2- c 3 ) + ( c 1 + c 2 ) x + c 3 x 2 . We must find a nontrivial solution for c 1- c 2- c 3 = 0 c 1 + c 2 = 0 c 3 = 0 . However, it is easy to see that c 1 = c 2 = c 3 = 0, and the vectors are linearly independent. 1 T F (f) The determinant of A = 1 1000 2 3 4 5 6 1000 7 8 1000 9 8 7 6 5 4 3 2 1000 1 2 3 1000 4 is positive....

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