340LExam3B - 1 1 1 and 1-3 2 and let y = 1-4-3 Write y as...

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M340L EXAM 3B 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 6 × 6 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 = " - 5 4 - 12 9 # . (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 28. Show that 28 is an eigenvalue of A by finding an eigenvector v for λ = 28, and explain why your v is in fact an eigenvector for λ = 28 .
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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. (10 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 fit a parabola y = β + β 1 x + β 2 x 2 to the data (0 , 2) , (1 , 6) , (3 , 26) , (4 , 50). Give a matrix X and a vector y so that the equation X β β 1 β 2 = y leads to a least-squares fit of the parabola with the given data. 7. (8 points) Find a least-squares solution of 1 5 3 1-2 4 x = 2-1 5 ....
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340LExam3B - 1 1 1 and 1-3 2 and let y = 1-4-3 Write y as...

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