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# exam3 - 18.06 QUIZ 3 Your PRINTED name is SOLUTIONS Please...

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18.06 QUIZ 3 May 07, 2007 Grading 1 2 3 4 Total: Your PRINTED name is: SOLUTIONS Please circle your recitation: (1) M 2 2-131 A. Osorno (2) M 3 2-131 A. Osorno (3) M 3 2-132 A. Pissarra Pires (4) T 11 2-132 K. Meszaros (5) T 12 2-132 K. Meszaros (6) T 1 2-132 Jerin Gu (7) T 2 2-132 Jerin Gu

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Problem 1 (25 points) (a) Compute the singular value decomposition A = U Σ V T for A = 0 1 - 1 1 1 0 . (b) Find orthonormal bases for all four fundamental subspaces of A . Solution 1 (a) A T A = 2 - 1 - 1 2 , with eigenvalues 3 and 1. Thus σ 1 = 3 and σ 2 = 1 . v 1 is a normal eigenvector corresponding to 3, so v 1 = 1 / 2 - 1 / 2 . v 2 is a normal eigenvector corresponding to 1, so v 2 = 1 / 2 1 / 2 . u 1 = 1 3 Av 1 = - 1 / 6 - 2 / 6 1 / 6 ; u 2 = Av 2 = 1 / 2 0 1 / 2 . For u 3 we nd a basis for the nullspace of A T : u 3 = - 1 / 3 1 / 3 1 / 3 . Thus
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