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# 2005stt - University of Toronto at Scarborough Department...

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University of Toronto at Scarborough Midterm Test MATA23 – Linear Algebra I Examiner: N. Cheredeko Date: February 11, 2005 Duration: 110 minutes 1. [7 points] (a) Give the deﬁnition of the span of n vectors. (b) What is the dimension of sp ± [1 , - 2 , 0] , [ - 2 , 4 , 1] , [ - 1 , - 2 , 1] , [ - 3 , 2 , 2] ² ? 2. [5 points] Use the vector method to show that midpoint of P 1 ( x 1 , y 1 ) and P 2 ( x 2 , y 2 ) is P ³ x 1 + x 2 2 , y 1 + y 2 2 ´ . 3. [3 points] Show that µ µ ¯ u + ¯ v µ µ 2 + µ µ u - v µ µ 2 = 2( k u k 2 + k v k 2 ). 4. [2 points] Can ¯ u · ¯ v = - 7 if k ¯ u k = 3 and k ¯ v k = 2? Defend your answer. 5. (a) [4 points] Find the projection of ¯ u = [2 , - 3 , 1] onto ¯ d = [1 , - 1 , 3] and express ¯ u = ¯ u 1 + ¯ u 2 where ¯ u 1 is parallel to ¯ d and ¯ u 2 is orthogonal to ¯ d . (b) [3 points] Calculate the distance from point (3 , 1 , 4) to the line x - 2 y + z = 0. (c)

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2005stt - University of Toronto at Scarborough Department...

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