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Exam3A - t = 1 2 t-3 t 2(b(6 points Show that T is a linear...

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M340L EXAM 3A SPRING, 2009 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. (10 points) Explain in detail why eigenvalues of a matrix A must be solutions of the equation det( A - λI ) = 0
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YOUR SCORE: /100 2. (10 points) Explain in detail why two vectors u and v in R p are orthogonal exactly when || u + v || 2 = || u || 2 + || v || 2 . 3. (10 points) Is 1 1 - 1 an eigenvector of the matrix 3 0 - 1 2 3 1 - 3 4 5 ? If so, find its corresponding eigenvalue.
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4. (10 points) Find the eigenvalues of the matrix A = 7 0 0 2 5 - 1 2 4 1 . 5. (10 points) The eigenvalues of the matrix A = 2 2 - 1 1 3 - 1 - 1 - 2 2 . are λ = 5 , 1. Diago- nalize A if possible and if not, explain why not.
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6. Let T : P 2 P 3 be the tranformation that maps a polynomial p ( t ) into the polynomial 2 p ( t ) + 3 t p ( t ). (a) (6 points) Find the image of
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Unformatted text preview: ( t ) = 1 + 2 t-3 t 2 . (b) (6 points) Show that T is a linear transformation. (c) (6 points) Find the matrix of T relative to the standard bases of P 2 and P 3 . 7. Let W be the subspace of R 4 spanned by the orthogonal vectors 3 2-1 4 and 2 2 6-1 , and let y = -5 2-5 9 . (a) (10 points) Write y as the sum of a vector in W and a vector in W ⊥ . (b) (6 points) Find the distance from y to the subspace W . 8. (8 points) Find an orthogonal basis for the column space of A = 2 7-8-1 1-14 1-1 8 4 1 . 9. (8 points) Find a least squares solution of A x = b , where A = -1 2 2-3-1 3 and b = 3 1 2 ....
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