linalg040April2007

# linalg040April2007 - Linear Algebra 040b Final Examination...

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Linear Algebra 040b Final Examination Saturday, April 14, 2007 1. [ 2 marks ] Let θ be the angle between the vectors (1 , - 2 , 3) and (3 , 3 , 1). Find cos( θ ). 2. [ 4 marks ] Let be the line in R 3 passing through the points (2 , 4 , 1) and (4 , 0 , 7). (a) Find parametric equations for . (b) Find the point of intersection of with the xy -plane. 3. [ 8 marks ] Find (if possible) the conditions on a, b and c for which the following system of equations has no solution, one solution and infinitely many solutions. x - y + 3 z = a y + z = b x + 4 z = c (Note: you do not have to find actual solutions.) 4. [ 8 marks ] Let W = Span( S ) where S = 1 2 0 1 , 2 3 1 3 , 0 1 2 2 , 1 4 1 2 . Find a subset of S which is a basis of W . 5. The characteristic polynomial of the matrix A = 0 - 2 - 3 1 3 3 0 0 1 is given by p ( λ ) = ( λ - 1) 2 ( λ - 2) .

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