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e3s-415 - Name MTH 415 Exam 3 April 1 2009 ID Number No...

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Name: ID Number: MTH 415 Exam 3 April 1, 2009 No calculators or any other devices are allowed on this exam. Read each question carefully. If any question is not clear, ask for clarification. Write your solutions clearly and legibly; no credit will be given for illegible solutions. If you present different answers for the same problem, the worst answer will be graded. Answer each question completely, and show all your work. 1. (22 points) Consider the matrices A = 1 3 - 3 2 - 2 10 3 1 7 and B = 1 5 - 3 0 - 6 6 1 - 1 3 . (a) Is N ( A ) = N ( B )? Justify your answer. (b) Is R ( A ) = R ( B )? Justify your answer. Part (a): We now that N ( A ) = N ( B ) iff E A = E B . A = 1 3 - 3 2 - 2 10 3 1 7 1 3 - 3 0 - 8 16 0 - 8 16 1 0 3 0 1 - 2 0 0 0 = E A , B = 1 5 - 3 0 - 6 6 1 - 1 3 1 5 - 3 0 - 6 6 0 - 6 6 1 0 2 0 1 - 1 0 0 0 = E B . Since E A = E B , we conclude N ( A ) = N ( B ) . Part (b): We now that R ( A ) = R ( B ) iff E A T = E B T . A T = 1 2 3 3 - 2 1 - 3 10 7 1 2 3 0 - 8 - 8 0 16 16 1 0 1 0 1 1 0 0 0 = E A T , B T = 1 0 1 5 - 6 - 1 - 3 6 3 1 0 1 0 - 6 - 6 0 6 6 1 0 1 0 1 1 0 0 0 = E B T . Since E A T = E B T , we conclude R ( A ) = R ( B ) . # Score 1 2 3 4 5 Σ 1
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2. (18 points) Find a basis and state the dimension of the vector space of all skew-symmetric 3 × 3 real matrices. We are interested in finding a basis of the space of 3 × 3 skew-symmetric matrices, that is, SS (3 , 3) = ' A M (3 , 3) : A = - A T .
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