1502CTest3

# 1502CTest3 - P = 1 3 5 ˆ ˜ ˜ ˜ Q = 2 7 1 ˆ ˜ ˜ ˜ R...

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Name±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± 15 April 2004 Teaching Assistant±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± Hour Test 3 Math 1502C 10am Andrew Instructions: 1. Closed book. 2. Show your work and explain your answers and reasoning. 3. Calculators may be used, but pay particular attention to instruction 2. To receive credit, you must show your work. Unexplained answers, and answers not supported by the work you show, will not receive credit. 4. Express your answers in simplified form. 1. (25) Use the Normal Equations method to find the least squares solution to Ax = b where A = 1 1 0 2 2 - 1 ˆ ˜ ˜ ˜ and b = - 1 16 3 ˆ ˜ ˜ ˜ . 2. (25) a. Are the vectors 1 - 4 - 1 - 1 ˆ ˜ ˜ ˜ ˜ , 3 0 1 1 ˆ ˜ ˜ ˜ ˜ , 1 2 1 1 ˆ ˜ ˜ ˜ ˜ , 4 2 0 0 ˆ ˜ ˜ ˜ ˜ linearly independent? b. What is the dimension of the subspace of R 4 that they span? 3. (25) Use the cross product to find an equation for the plane containing

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Unformatted text preview: P = 1 3 5 ˆ ˜ ˜ ˜ , Q = 2 7 1 ˆ ˜ ˜ ˜ , R = 1-3-2 ˆ ˜ ˜ ˜ . 4. (25) The matrix A = 1 1-5 5 3 3-2 ˆ ˜ ˜ ˜ has characteristic polynomial 1 -t ( ) 2 -t ( ) 2 . a. Find all eigenvalues and eigenvectors of A . b. Either find an invertible matrix S and a diagonal matrix D with S-1 AS = D or explain why no such S and D exist. Name±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± Page 2 of 2 Math 1502C 10am Andrew Hour Test 3 Teaching Assistant±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± 15 April 2004 Answers. 1. 2 5 ˆ ˜ 2. a. No, they are linearly dependent. b. 3 3. -52 x + 7 y-6 z =-61 4. a. l = 1 , -1-2 1 ˆ ˜ ˜ ˜ and = 2 , -1 1 ˆ ˜ ˜ ˜ b. Since the eigenvalue 2 does not have 2 linearly independent eigenvalues, no such S and D exist. A is not diagonalizable....
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1502CTest3 - P = 1 3 5 ˆ ˜ ˜ ˜ Q = 2 7 1 ˆ ˜ ˜ ˜ R...

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