223midterm220111

# 223midterm220111 - A . 3 of 7 3. Consider the susbpace W =...

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University of Toronto Department of Mathematics MAT223H1S Linear Algebra I Midterm Examination II March 24, 2011 M. Czubak, A. Macleod, B. Smithling, S. Uppal Duration: 1 hour 30 minutes Last Name: Given Name: Student Number: Tutorial Code: No calculators or other aids are allowed. FOR MARKER USE ONLY Question Mark 1 /10 2 /10 3 /10 4 /10 5 /5 6 /5 TOTAL /50 1 of 7

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1. Let L be the line in R 3 that passes through the points (2 , 1 , 3) and (5 , 0 , 1). Let P be the point (3 , 2 , - 1). (a) Find the shortest distance from the point P to the line L and ﬁnd the point Q on the line closest to the point P . (b) Find the equation of the plane that contains the point P and the line L . 2 of 7
2. Let A = 2 - 4 6 8 2 - 1 3 2 4 - 5 9 10 0 - 1 1 2 . Find bases for the row and column spaces of A and deter- mine the rank of

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Unformatted text preview: A . 3 of 7 3. Consider the susbpace W = span { (1 , 3 , 1 ,-1) , (2 , 6 , , 1) } of R 4 . Find an orthonormal basis for W ⊥ (the orthogonal complement of W ). 4 of 7 4. Find a basis for, and state the dimension of the subspace W = { A ∈ M 22 | A ± 1 1-1 0 ² = ± 1-1 1 ² A } of M 22 . 5 of 7 5. Find a basis for P 3 ( R ) that contains the polynomials x 3-2 x 2 +1 and 2 x 3 +4 x 2-x +3. Justify your answer. 6 of 7 6. Let A be an m × n matrix and let W be the nullspace of A . Prove that for all X ∈ R n , AX = A ( proj W ⊥ ( X )) . where proj W ⊥ ( X ) denotes the orthogonal projection of X onto W ⊥ . 7 of 7...
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## This note was uploaded on 04/09/2012 for the course MATH 223 taught by Professor P during the Spring '11 term at University of Toronto.

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223midterm220111 - A . 3 of 7 3. Consider the susbpace W =...

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