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223midterm20099

# 223midterm20099 - University of Toronto Department of...

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University of Toronto Department of Mathematics MAT223H1F Linear Algebra I Midterm Examination October 22, 2009 H. Kim, S. Kudla, F. Murnaghan, S. Uppal Duration: 1 hour 50 minutes Last Name: Given Name: Student Number: Tutorial Code: No calculators or other aids are allowed. FOR MARKER USE ONLY Question Mark 1 /6 2 /10 3 /10 4 /10 5 /10 6 /10 7 /4 8 /5 TOTAL /65 1 of 9

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[6] 1. Find all solutions of the homogeneous linear system Ax = 0, where A is a 3 × 6 matrix whose reduced row echelon form is 1 2 0 3 0 5 0 0 1 4 0 6 0 0 0 0 1 1 . Express your answer in parametric form. 2 of 9
[8] 2. Find a 2 × 2 matrix A such that A 1 2 - 1 0 = 3 1 2 3 . 3 of 9

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[10] 3. Let A = 3 4 5 3 1 1 1 1 1 2 3 1 3 5 7 3 . Find a non-singular (invertible) matrix U such that R = UA , where R is the reduced row echelon form of A . 4 of 9
[10] 4. Let P 4 ( R ) be the vector space of all real valued polynomials of degree less than or equal to four. Let W be the subspace of P 4 ( R ) given by W = { p ( x ) P 4 ( R ) | p ( - 2) = p (2) } . Find a finite subset S of W such that W = span ( S ). 5 of 9

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5. Let W be the subspace of R 4 defined by W = Span { (1 , 0 , 1 , 0) , (1 , 1 , 1 , 1) , (0 , 1 , 1 , 1) } , and let S be the subset of R 4 given by S = { (1 , 0 , 1 , 0) , (0 , 1 , 0 , 1) , (2 , 3 , 2 , 3) } . [5] (a) Show S W . [5] (b) Does span

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223midterm20099 - University of Toronto Department of...

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