Exam3A - 1 1 1 and -1 3-2 , and let y = -1 4 3 . Write y as...

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M340L EXAM 3A SPRING, 2008 Dr. Schurle Your name: Your UTEID: Show all your work on these pages. Be organized and neat. Your work should be your own; there should be no talking, reading notes, checking laptops, using cellphones, . . . . 1. (8 points) A is a 5 × 5 matrix with 3 eigenvalues. Two eigenspaces each have dimension 2. Is it possible that A is not diagonalizable? Justify your answer.
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2. (a) (8 points) Find the eigenvalues of the matrix A = " - 2 2 - 10 7 # . (b) (8 points) Find eigenvectors corresponding to the eigenvalues you found in part (a).
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(c)(4 points) Continuing with the problem from the previous page, give a diagonal matrix D and an invertible matrix P such that A = PDP - 1 . 3. (8 points) Suppose that A is a 6 × 6 matrix for which the sum of each row is 17. Show that 17 is an eigenvalue of A by finding an eigenvector v for λ = 17, and explain why your v is in fact an eigenvector for λ = 17 .
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4. (8 points) Let W be the subspace of R 3 spanned by the orthogonal vectors
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Unformatted text preview: 1 1 1 and -1 3-2 , and let y = -1 4 3 . Write y as the sum of a vector in W and a vector orthogonal to W . 5. (8 points) Find an orthogonal basis for the column space of A = -1 2 2 1 1 1 . Then write down a matrix Q such that A = QR , where R is an invertible upper triangular matrix. Finally, write down a formula for R involving Q and A but DO NOT CALCULATE R. 6. (8 points) Suppose we want to t a parabola y = + 1 x + 2 x 2 to the data (0 , 1) , (1 , 5) , (2 , 14) , (3 , 25). Give a matrix X and a vector y so that the equation X 1 2 = y leads to a least-squares t of the parabola with the given data. 7. (8 points) Find a least-squares solution of 1 5 3 1-2 4 x = 4-2-3 ....
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Exam3A - 1 1 1 and -1 3-2 , and let y = -1 4 3 . Write y as...

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