quiz2_S07_solutions

quiz2_S07_solutions - Princeton University Department of...

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Princeton University Department of Mathematics MAT 202 – Linear Algebra with Applications QUIZ 2 – Spring 2007 SOLUTIONS (1) Let A = 0 1 1 1 1 1 and b = 2 3 5 . (a) Find the least-squares solution of the system Ax = b . (b) What is the vector in the image of A which is closest to the vector b ? Solution: (a) Since A T A = » 0 1 1 1 1 1 2 4 0 1 1 1 1 1 3 5 = » 2 2 2 3 is invertible and A T b = » 0 1 1 1 1 1 2 4 2 3 5 3 5 = » 8 10 the least-squares solution of the system Ax = b is x * = ( A T A ) - 1 A T b = 1 2 » 3 - 2 - 2 2 – » 8 10 = » 2 2 . (b) The vector in the image of A which is closest to the vector b is Ax * = 2 4 0 1 1 1 1 1 3 5 » 2 2 = 2 4 2 4 4 3 5 . (2) Let A = 1 1 0 1 0 1 0 1 0 0 0 1 . (a) Find the 3-volume of the parallelepiped defined by the columns of A . (b) Let v be a unit vector in R 4 . The augmented matrix [ A | v ] is 4 × 4 . What is the maximum value det[ A | v ] may take? Solution: (a) We have that det( A T A ) = det 0 B B @ 2 4 1 1 0 0 1 0 1 0 0 1 0 1 3 5 2 6 6 4 1 1 0 1 0 1 0 1 0 0 0 1 3 7 7 5 1 C C A = det
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