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Unformatted text preview: MATH 202 MIDTERM EXAM Wednesday, March 9, 2005, 7:30PM9:00PM The average on this exam was 59 percent. 1. (16 points) Consider A = 1 1 2 2 1 3 1 1 1 . (a) Compute A 1 . A 1 = 2 3 5 1 1 1 1 2 3 (b) If B is a 3 × 3 matrix such that ABA = I , then is B necessarily invertible? (Justify your answer.) Yes. Multiply by A 1 on the left and on the right to obtain B = A 1 I · A 1 = A 2 , so B 1 = A 2 . (c) If ~x = 3 2 6 in standard coordinates, then what are the coordinates of ~x with respect to the basis B = 1 2 1 ,  1 1 1 , 2 3 1 A is the matrix that converts Bcoordinates to standard coordinates, since its i th column is the same as the i th basis vector. Thus the matrix that converts standard coordinates to Bcoordinates is A 1 from part (a). So 2 3 5 1 1 1 1 2 3 3 2 6 = 30 7 17 are the standard coordinates of ~x . 2. (20 points) Consider the matrix A = 1 1 5 3 1 3 2 1 2 1 7 4 2 1 2 1 which reduces to rref ( A ) = 1 2 1 1 3 2 1 (a) Find a basis for the kernel of A .  2 3 1 ,  1 2 1 (b) Find a basis for the image of A ....
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 Spring '08
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 Math, Linear Algebra, Algebra, basis, 59 percent

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