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Exam3B - M340L EXAM 3B SPRING 2010 Dr Schurle Your name...

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M340L EXAM 3B 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 - 2 1 3 - 8 2 2 7 - 1 2 ? If so, what is its eigenvalue? If not, explain why not.
4. (18 points) Determine whether the matrix A = 2 6 5 0 2 0 4 5 1 is diagonalizable. If it is, ﬁnd the relevant P and D . If it is not, explain why not.

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Exam3B - M340L EXAM 3B SPRING 2010 Dr Schurle Your name...

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