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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 4by6 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 λI ) = 1 λ 3 3 3 λ = (1 λ )( 3 λ ) 9 = λ 2 +2 λ 12 = 0 . This gives λ = 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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This note was uploaded on 04/23/2008 for the course MATH 20F taught by Professor Buss during the Spring '03 term at UCSD.
 Spring '03
 BUSS
 Math, Linear Algebra, Algebra

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