ef08k - MAS 3105 AM: April 24, 2008 Final Exam and Key...

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Unformatted text preview: MAS 3105 AM: April 24, 2008 Final Exam and Key Prof. S. Hudson 1) (10pts) Find e A Compute each entry; you can leave those in unsimplified forms, such as e + + 3 6 : A = 1- 1 1- 4 = 4 1 1 15 4 4 1 1 1 4- 1- 15 4- 1 1 2) (10pts) For the matrix A in problem 1), explain why A 2 = A must be true, without actually multiplying out A * A . Then, find a matrix B 6 = A so that B 2 = A . 3) (15 pts) a) Find the least squares solution to A x = b , using this QR factorization: A = 1- 2- 1 2 1 2- 4 2 4 = 1 5 1- 2- 4 2 1 2 2- 4 2 4 2- 1 5- 2 1 4- 1 2 where b = - 1 1 1- 2 3b) Let S = R ( A ). Find proj S b . 4) (10pts) Find the standard matrix representation for the following linear transformation: L reflects each vector about the line x 1 = x 2 , and then projects it onto the x 1 axis. 5) (10pts) Show that the product of two unitary matrices is unitary. [Hint - use the formula with U H .] 6) [15pts] Factor A into a product XDX- 1 where D is diagonal. A = 5 6- 2- 2 7) (20 points) Answer True or False. Assume the matrices are 3x3 (if necessary)....
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ef08k - MAS 3105 AM: April 24, 2008 Final Exam and Key...

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