223midterm220119 - University of Toronto Department of...

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University of Toronto Department of Mathematics MAT223H1F Linear Algebra I Midterm Exam II November 24, 2011 C. Anghel, S. Arkhipov, S. Shahroki-Tehrani, S. Uppal Duration: 1 hour 30 minutes Last Name: Given Name: Student Number: Tutorial Group: No calculators or other aids are allowed. FOR MARKER USE ONLY Question Mark 1 /10 2 /10 3 /10 4 /10 5 /10 6 /5 TOTAL /55 1 of 10
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1. Consider the points P = (1 , 2 , 3) and Q = (4 , 5 , 6) in R 3 . [6] (a) Find an equation for the plane that passes through the origin and contains the points P and Q . Express your answer in the form ax + by + cz = d . [2] (b) Find the orthogonal projection of ( - 1 , 0 , 8) onto the normal vector to the plane in part (a). [2] (c) Find the distance from the point ( - 1 , 0 , 8) to the plane in part (a). 2 of 10
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2. Consider the vectors v 1 = (1 , 1 , 0), and v 2 = (1 , 1 , 1) in R 3 . [5] (a) Find an orthonormal basis for the subspace W = Span { v 1 ,v 2 } of R 3 . [5] (b) Find the standard matrix of the orthogonal projection onto W . 4 of 10
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[10] 3. Let P 3 ( R ) be the vector space of real polynomials of degree at most 3 with inner product < a 0 + a 1 x + a 2 x
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223midterm220119 - University of Toronto Department of...

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