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# 2006wtt-a23 - University of Toronto at Scarborough...

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University of Toronto at Scarborough Department of Computer & Mathematical Sciences Midterm Test MATA23H3 Linear Algebra I Examiner: N. Cheredeko Date: February 10, 2006 Duration: 110 minutes 1. [10 points] (a) Determine whether - 1 3 2 is a linear combination of 2 0 1 and 0 2 4 . (b) Do 2 0 1 and 0 2 4 span R 3 ? Justify your answer. 2. [10 points] (a) Find the projection of w = 1 1 1 on the normal to the plane π with equation x - 7 y + 4 z = 0. (b) Find the distance from the point P (1 , 1 . 1) to the plane π . 3. [15 points] Let u = 3 4 0 and v = 0 0 3 . (a) Find the angle between u and v . (b) State and verify the Cauchy-Schwartz inequality for u and v . (c) Find a unit vector that is perpendicular to both u and v .

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MATA23H page 2 4. [15points] (a) Suppose A and B are invertible matrices and X is a matrix such that AXB = A + B . What is X ? (b) Suppose A and B are square matrices that commute. Prove that ( AB ) T = A T B T .
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