E3s-415 - 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

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Unformatted text preview: 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- 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- 8 16- 8 16 → 1 3 1- 2 = E A , B = 1 5- 3- 6 6 1- 1 3 → 1 5- 3- 6 6- 6 6 → 1 2 1- 1 = 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- 8- 8 16 16 → 1 1 1 1 = E A T , B T = 1 1 5- 6- 1- 3 6 3 → 1 1- 6- 6 6 6 → 1 1 1 1 = E B T . Since E A T = E B T , we conclude R ( A ) = R ( B ) . # Score 1 2 3 4 5 Σ 1 2. (18 points) Find a basis and state the dimension of the vector space of all skew-symmetric 3 × 3 real matrices....
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This note was uploaded on 10/28/2010 for the course MATH Math 415 taught by Professor Dr.gabrielnagy during the Spring '09 term at Michigan State University.

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E3s-415 - 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

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