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Exam3A - M340L EXAM 3A SPRING 2010 Dr Schurle Your name...

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M340L EXAM 3A Your name: SPRING, 2010 Dr. Schurle 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) Suppose the matrix A has eigenvalues 3 and 5 with corresponding eigen- vectors v 1 and v 2 . Using only algebra and definitions, show why v 1 and v 2 are linearly independent.
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YOUR SCORE: /100 2. (10 points) Suppose the columns of a matrix U are orthonormal. Explain why U T U = I , where I is the identity matrix of the appropriate size, and then why ( U x ) · ( U y ) = x · y for vectors x and y of the appropriate size. 3. (8 points) Is v = 1 2 - 1 an eigenvector of the matrix 0 2 2 12 - 3 2 - 3 1 1 ? If so, what is its eigenvalue? If not, explain why not.
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4. (18 points) Determine whether the matrix A = 2 8 3 0 3 0 4 2 1 is diagonalizable. If it is, find the relevant P and D . If it is not, explain why not.
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5. Remember that P 2 is the vector space of all polynomials of degree no more than 2.
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