2009btest3web - is worth 5 points. You do NOT need to...

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M346 Third Midterm Exam, November 20, 2009 1) Gram Schmidt: a)(10 points) On R 3 with the usual inner product, Use Gram-Schmidt to convert x 1 = (1 , 2 , 0) T , x 2 = (3 , 1 , 1) T , x 3 = (4 , 3 , 5) T to an orthogonal basis. b)(15 points) On R 2 [ t ] with the inner product a f | g A = i 2 0 f ( t ) g ( t ) dt , trans- form { 1 , t, t 2 } to an orthogonal basis. 2. a)(15 points) Find the equation of the best line through the points (1 , 4), (2 , 1), and (3 , 2). b)(10 points) Let V be the subspace of R 3 that is the span of the vectors (1 , 2 , 3) T and (1 , 1 , 1) T . Find the point in V that is closest to ( 4 , 1 , 2) T . 3. On C 3 with the usual inner product, let L x 1 x 2 x 3 = x 1 + ix 2 ix 3 2 x 2 + (1 i ) x 3 ix 1 + 3 x 2 + x 3 a)(5 points) Find the matrix of L b)(10 points) Let x = 1 10 100 . Compute L ( x ). c)(10 points) Let V be the space of real-valued functions on the real line, with the inner product a f | g A = i −∞ f ( t ) g ( t ) dt . Let A : V V be the linear transformation A = t + d/dt (That is, ( A ( f ))( t ) = tf ( t ) + f ( t )). Let g ( t ) = e t 2 / 2 . Compute Ag and A g . 4. Grab bag. These are short-answer or true/false questions. Each question
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Unformatted text preview: is worth 5 points. You do NOT need to justify your answers, and partial credit will NOT be given. a) True or false? The matrix p 5 4 i − 4 i − 1 P has orthogonal eigenvectors. b) True or false? The matrix 1 √ 7 p 2 − i − 1 + i 1 + i 2 + i P is unitary. c) Let x ( t ) be the solution to d x dt = A x , where A = 1 2 3 − 1 − 1 4 − 2 1 5 − 3 − 4 − 5 and x (0) = (5 , − 3 , 1 , 1) T Find the limit, as t → ∞ , of | x ( t ) | . (This has a quick 1 and easy solution, and you do NOT have to diagonalize A !) d) True or false? If a matrix M satisFes M = M T , then the eigenvalues of M are real. e) True or false? If a matrix is unitary, then it is not Hermitian. 2...
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This note was uploaded on 04/10/2011 for the course M 346 taught by Professor Radin during the Spring '08 term at University of Texas.

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2009btest3web - is worth 5 points. You do NOT need to...

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