223midterm20089sol - University of Toronto Department of...

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Unformatted text preview: University of Toronto Department of Mathematics MAT223H1F Linear Algebra I Midterm Examination October 23, 2008 G. Gotsbacher, B. Jacob, S. Uppal, F. Ziltener 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 /8 2 /8 3 /8 4 /10 5 /10 6 /10 7 /6 TOTAL /60 1 of 9 For each of the statements below, decide if it is true or false. Indicate your answer by shading in the box corresponding to your choice. Justify your answer by providing an appropriate proof or counter example. [4] 1(a) The set S = { ( x,y ) ∈ R 2 | xy = 0 } is a vector subspace of R 2 . true false The statement is false: both (1 , 0) and (0 , 1) are in S , but (1 , 0) + (0 , 1) = (1 , 1) is not in S . [4] 1(b) The set S = { (1 , , , 0) , (0 , 1 , 1 , 0) , (0 , , 1 , 1) , (1 , , , 1) , (- 1 , 1 , 2 , 1) } spans R 4 . true false The statement is true. We’ll give two different solutions. First solution We’ll check that given any vector ( a,b,c,d ) ∈ R 4 there exist real numbers x,y,z,t,w such that x (1 , , , 0) + y (0 , 1 , 1 , 0) + z (0 , , 1 , 1) + t (1 , , , 1) + w (- 1 , 1 , 2 , 1) = ( a,b,c,d ) So we just need to check that the corresponding linear system (in variables x,y,z,t,w ) is consistent for any given choice of a,b,c,d . Using Gaussian elimination we get 1 0 0 1- 1 a 0 1 0 0 1 b 0 1 1 0 2 c 0 0 1 1 1 d R 3 7→ R 3- R 2-------→ 1 0 0 1- 1 a 0 1 0 0 1 b 0 0 1 0 1 c- b 0 0 1 1 1 d R 4 7→ R 4- R 3-------→ 1 0 0 1- 1 a 0 1 0 0 1 b 0 0 1 0 1 c- b 0 0 0 1 d- c + b and the last system is clearly consistent for any value of a,b,c,d ∈ R ....
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This note was uploaded on 06/14/2010 for the course MATH MAT223 taught by Professor Uppal during the Fall '08 term at University of Toronto.

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223midterm20089sol - University of Toronto Department of...

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