midterm_S05_solutions

midterm_S05_solutions - MATH 202- MIDTERM EXAM Wednesday,...

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Unformatted text preview: MATH 202- MIDTERM EXAM Wednesday, March 9, 2005, 7:30PM-9:00PM 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 B-coordinates 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 B-coordinates 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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midterm_S05_solutions - MATH 202- MIDTERM EXAM Wednesday,...

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