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# ef04k - MAS 3105 Final Exam and Key Prof S Hudson 1(10pts...

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MAS 3105 April 19, 2004 Final Exam and Key Prof. S. Hudson 1) (10pts) Suppose that A is a square singular matrix. What can you say about the product A adj A ? 2) (10 pts) a) Find the least squares solution to A x = b , using this QR factorization: A = 1 - 2 1 2 0 1 2 - 4 2 4 0 0 = 1 5 1 - 2 - 4 2 1 2 2 - 4 2 4 2 - 1 5 - 2 1 0 4 - 1 0 0 2 where b = - 1 1 1 - 2 2b) Let S = R ( A ). Find proj S b . 3) (15pts) a) Given that A is row equivalent to U , find a basis for N ( A ). Check ! A = 1 - 2 1 1 2 - 1 3 0 2 - 2 0 1 1 3 4 1 2 5 13 5 U = 1 - 2 1 1 2 0 1 1 3 0 0 0 0 0 1 0 0 0 0 0 3b) Find a basis of the row space of A . 3c) Find a basis of the column space of A . 4) (10pts) Find the standard matrix representation for each of the following linear trans- formations, a) L rotates each vector x in R 2 by 45 in the clockwise direction. b) L reflects each vector about the line x 1 = x 2 , and then projects it onto the x 1 axis.

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ef04k - MAS 3105 Final Exam and Key Prof S Hudson 1(10pts...

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