Spring 2007 - Takeda's Class - Final Exam

# Spring 2007 - Takeda's Class - Final Exam - Math 20F, Final...

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Unformatted text preview: Math 20F, Final Exam June 11, 2007 Name : PID : TA : Sec. No : Sec. Time : This exam consists of 13 pages including this front page. The last two pages are extra credit problems. Ground Rules 1. No calculator is allowed. 2. Show your work for every problem. A correct answer without any justification will receive no credit. 3. You may use two 4-by-6 index cards, both sides. 4. You have two hours for this exam. Score 1 10 2 10 3 12 4 12 5 10 6 10 7 10 8 6 9 10 10 10 Total 100 1 1. (a) Determine if 1 3- 4 is in Nul A , where A = 3- 5- 3 6- 2- 8 4 1 . Answer : 3- 5- 3 6- 2- 8 4 1 1 3- 4 = 3- 15 + 12 6- 6 + 0- 8 + 12- 4 = . Hence it is in Nul A . (b) Let x = 2- 3 2 and y = 1 1 . Find proj y x . Answer : proj y x = x y y y y = 2 + 2 1 + 1 1 1 = 2 2 2 2. (a) Compute 1 3- 1 2 2 6- 2 4 1 0 2- 1 0 7 2 2 . Answer : Notice that the second row is 2 times the first row. Hence 1 3- 1 2 2 6- 2 4 1 0 2- 1 0 7 2 2 = 0 . (b) Find rank A , where A = 2 3- 1- 8- 7 6 6- 1- 7 Answer : 2 3- 1- 8- 7 6 6- 1- 7 2 3- 1 5 2- 10- 4 2 3- 1 0 5 2 0 0 . Hence only two of the columns have a pivot, and so rank A = 2. 3 3. (a) Find all the eigenvalues of the matrix A = 2- 2 8 5- 2- 4 3 . Answer : det( A- I ) = 2- - 2 8 5- - 2- 4 3- = (2- ) 5- - 2- 4 3- = (2- )( 2- 8 + 7) = (2- )( - 7)( - 1) ....
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## Spring 2007 - Takeda's Class - Final Exam - Math 20F, Final...

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