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exam1solns - Midterm Exam Math 20F Name Section Read all of...

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Midterm Exam Math 20F Name: 8/22/08 Section: Read all of the following information before starting the exam: READ EACH OF THE PROBLEMS OF THE EXAM CAREFULLY! Show all work, clearly and in order, if you want to get full credit. I reserve the right to take off points if I cannot see how you arrived at your answer (even if your final answer is correct). A single 8 1/2 × 11 sheet of notes (double sided) is allowed. No calculators are permitted. Circle or otherwise indicate your final answers. Please keep your written answers clear, concise and to the poin. This test has 5 problems and is worth 100 points.. It is your responsibility to make sure that you have all of the pages! Turn off cellphones, etc. Good luck! 1 2 3 4 5
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1. ( 25 points ) a. ( 5 pts ) Solve the system of equations: 2 x 1 + 0 x 2 + 3 x 3 = 5 x 1 + x 2 + x 3 = 2 x 1 - 2 x 2 + 2 x 3 = 4 Answer: Note (swapping equations 1 & 2 to make things a bit cleaner) we get 1 1 1 2 2 0 3 5 1 - 2 2 4 1 1 1 2 0 - 2 1 1 0 - 3 1 2 1 1 1 2 0 1 0 - 1 0 - 3 1 2 1 1 1 2 0 1 0 - 1 0 0 1 - 1 1 0 0 4 0 1 0 - 1 0 0 1 - 1 . so x = 4 - 1 - 1 . b. ( 10 pts ) Find the matrix of the linear transformation T : R 2 R 2 that rotates points about the origin π/ 2 radians clockwise . Compute T 2 1 . Answer: Note: T 1 0 = 0 - 1 and T 0 1 = 1 0 so A = 0 1 - 1 0 . Also, T 2 1 = 0 1 - 1 0 2 1 = 1 - 2 . c. ( 10 pts ) Let A = 1 2 3 1 2 3 1 4 . Find the solution set for A x = 0. In other words, find Nul( A ). Answer Note 1 2 3 1 2 3 1 4 1 2 3 1 0 - 1 - 5 2 1 0 - 7 5 0 1 5 - 2 .
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