Fall 2005 - Nagy's Class - Final Exam

Fall 2005 - Nagy's Class - Final Exam - Print Name: Student...

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Print Name: Student Number: Section Time: Math 20F. Final Exam December 7, 2005 Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly and legibly. No credit will be given for illegible solutions. 1. Consider the matrix A = 1 - 1 0 2 0 1 0 1 - 1 . (a) (4 Pts.) Find the inverse of the matrix A . (b) (2 Pts.) Use the part (1a) to solve the system A x = b , with b = 1 - 1 3 . (a) 1 - 1 0 | 1 0 0 2 0 1 | 0 1 0 0 1 - 1 | 0 0 1 1 - 1 0 | 1 0 0 0 2 1 | - 2 1 0 0 1 - 1 | 0 0 1 2 - 2 0 | 2 0 0 0 2 1 | - 2 1 0 0 2 - 2 | 0 0 2 , 2 0 1 | 0 1 0 0 2 1 | - 2 1 0 0 0 - 3 | 2 - 1 2 6 0 3 | 0 3 0 0 6 3 | - 6 3 0 0 0 3 | - 2 1 - 2 6 0 0 | 2 2 2 0 6 0 | - 4 2 2 0 0 3 | - 2 1 - 2 , 6 0 0 | 2 2 2 0 6 0 | - 4 2 2 0 0 6 | - 4 2 - 4 , A - 1 = 1 3 1 1 1 - 2 1 1 - 2 1 - 2 . (b) x = A - 1 x = 1 3 1 1 1 - 2 1 1 - 2 1 - 2 1 - 1 3 = 1 3 0 + 3 - 3 + 3 - 3 - 6 = 1 0 - 3 . So the answer is x = 1 0 - 3 .
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2. Let T : IR 2 IR 3 be a linear transformation given by T ( x 1 , x 2 ) = x 1 - 2 x 2 3 x 1 + x 2 x 2 . (a) (2 Pts.) Find the matrix A associated to the linear transformation T using the standard bases in IR 3 and IR 2 . (b) (2 Pts.) Is T one-to-one? Justify your answer. (c) (2 Pts.) Is T onto? Justify your answer. (a) T (1 , 0) = 1 3 0 , T (0 , 1) = - 2 1 1 , A = [ T ( e 1 ) , T ( e 2 )] = 1 - 2 3 1 0 1 . (b) T one-to-one N ( A ) = { 0 } , and x N ( A ) A x = 0 , 1 - 2 3 1 0 1 1 - 2 0 7 0 1 1 0 0 1 0 0 . Then,
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Fall 2005 - Nagy's Class - Final Exam - Print Name: Student...

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