Math416_Midterm1_sol

# Math416_Midterm1_sol - Math 416 Abstract Linear Algebra...

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Math 416 - Abstract Linear Algebra Fall 2011, section E1 Midterm 1, September 21 Name: Solutions No calculators, electronic devices, books, or notes may be used. Show your work. No credit for answers without justiﬁcation. Good luck! 1. /10 2. /10 3. /10 4. /10 Total: /40 1

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Problem 1a. (8 pts) Find all solutions (if any) of the system x 1 + 2 x 2 - x 3 + 2 x 4 = 3 3 x 1 + 7 x 2 + 5 x 4 = 8 - x 1 + 7 x 3 - 2 x 4 = - 1 . 1 2 - 1 2 3 3 7 0 5 8 - 1 0 7 - 2 - 1 1 2 - 1 2 3 0 1 3 - 1 - 1 0 2 6 0 2 1 2 - 1 2 3 0 1 3 - 1 - 1 0 0 0 2 4 1 2 - 1 2 3 0 1 3 - 1 - 1 0 0 0 1 2 1 2 - 1 0 - 1 0 1 3 0 1 0 0 0 1 2 1 0 - 7 0 - 3 0 1 3 0 1 0 0 0 1 2 The only free variable is x 3 . The solution set is { - 3 + 7 x 3 1 - 3 x 3 x 3 2 R 4 | x 3 R } = { - 3 1 0 2 + x 3 7 - 3 1 0 R 4 | x 3 R } . b. (2 pts) If we changed the right-hand side 3 8 - 1 to some other vector b R 3 , would the system necessarily have a solution? If so, justify; if not, ﬁnd an explicit right-hand side b R 3 such that the system has no solution.
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Math416_Midterm1_sol - Math 416 Abstract Linear Algebra...

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