Spring 2007 - Takeda's Class - Practice Final Exam

Spring 2007 - Takeda's Class - Practice Final Exam - Math...

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Unformatted text preview: Math 20F, Practice Final Exam Solutions June 6, 2007 Name : PID : TA : Sec. No : Sec. Time : This exam consists of 11 pages including this front page. 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 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 Total 100 1 1. (a) Compute- 2 5- 7 8 3 5 2 1 7 2 2 . Answer :- 2 5- 7 8 3 5 2 1 7 2 2 =- 2 3 0 0 5 2 1 7 2 2 =- 2 3 2 1 2 2 =- 12 . (b) Compute A- 1 , where A = 1 2- 4 0 1 2 0 0 1 Answer : 1 2- 4 1 0 0 0 1 2 0 1 0 0 0 1 0 0 1 1 2 0 1 0 4 0 1 0 0 1- 2 0 0 1 0 0 1 1 0 0 1- 2 8 0 1 0 0 1- 2 0 0 1 0 1 . Hence A- 1 = 1- 2 8 1- 2 1 . 2 2. (a) Find all the eigenvalues of the matrix A = 1- 3- 3- 3 . Answer : det( A- I ) = 1- - 3- 3- 3- = (1- )(- 3- )- 9 = 2 +2 - 12 = 0 . This gives =- 1 13 . (b) Find Nul A , where A is as above. Answer : Note that det A =- 12 6 = 0. Hence Nul A = { } . 3 3. (a) Let P 3 be the vector space of polynomials with degree less than or equal to 3. Let B = { 1 , 1+ t, 1+ t + t 2 , 1+ t + t 2 + t 3 } . Show that B is linearly independent. Answer : Let c 1 1 + c 2 (1 + t ) + c 3 (1 + t + t 2 ) + c 4 (1 + t + t 2 + t 3 ) = 0 ....
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Spring 2007 - Takeda's Class - Practice Final Exam - Math...

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