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Written2

# Written2 - Department of Mathematics and Statistics McGill...

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Department of Mathematics and Statistics, McGill University MATH 263 (Differential Equations and Linear Algebra) LINEAR ALGEBRA ASSIGNMENT amended This assignment will not be marked but is nevertheless essential to the course. Solutions will be provided. 1. Let A = 1 0 - 2 - 4 - 5 1 1 - 1 - 5 - 3 2 3 - 1 - 6 - 2 3 - 2 - 8 - 10 - 19 . (a) Find a basis for each of the following: the row space, the column space and the nullspace (ie kernel) of A . (b) Express each row of A in terms of your basis for the row space. (c) Express each column of A in terms of your basis for the column space. 2. Let U = { x ∈ R 3 | x 1 - 2 x 2 - x 3 = 0 } (a) Find the matrix R corresponding to reflection through U . (b) Find the matrices P and Q corresponding to orthogonal projection onto U and U respectively. (c) Express R in terms of P and Q . (d) Find A so that A x = 3 x for x U and A x = - 2 x for x U . (e) Express A in terms of P and Q . (f) Diagonalize the matrices P, Q, R, A.

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