quiz2_S07_solutions

quiz2_S07_solutions - Princeton University Department of...

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Unformatted text preview: Princeton University Department of Mathematics MAT 202 – Linear Algebra with Applications QUIZ 2 – Spring 2007 SOLUTIONS (1) Let A = 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 = » 1 1 1 1 1 – 2 4 1 1 1 1 1 3 5 = » 2 2 2 3 – is invertible and A T b = » 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 1 1 1 1 1 3 5 » 2 2 – = 2 4 2 4 4 3 5 . (2) Let A = 1 1 1 1 1 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 ....
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This note was uploaded on 04/28/2008 for the course MAT 202 taught by Professor Staff during the Spring '08 term at Princeton.

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quiz2_S07_solutions - Princeton University Department of...

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