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Unformatted text preview: MA 405003 Test 4 Spring 2008 Name Read each question carefully. You must SHOW ALL WORK for full credit. NO CALCULATORS. Good luck. 1. (20%) For the linear transformation T : R 3 R 2 defined by T x 1 x 2 x 3 = x 1 + x 2 x 3 Find the Range and Kernel of T . 2. (25%) For A = 1 1 2 2 (a) Find the eigenvalues of A . (b) Find the corresponding eigenspaces of A . (c) Find an invertible matrix P and a diagonal matrix D such that P 1 AP = D . (d) Find A 100 . (e) Find e A . 3. (10%) Determine the values of for which A = 0 0 0 1 1 0 1 1 is not diagonalizable. Justify. 4. (10%) Find the point on the line y = 2 x that is closest to the point (1 , 1). 5. (15%) Let v 1 = 1 3 (2 , 2 , 1 , 0) T and v 2 = 1 3 (0 , 1 , 2 , 2) T (a) Show { v 1 ,v 2 } is an orthonormal set. (b) Write v = (4 , , 10 , 8) T as a linear combination of v 1 and v 2 . Use dot products to solve, NOT a linear system....
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 Spring '08
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