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Spring 2005 - Nagy's Class - Exam 1

# Spring 2005 - Nagy's Class - Exam 1 - Math 20F Midterm...

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Math 20F Midterm Exam (version 1) April 29, 2005 1. (6 points) Consider the system x 1 - 2 x 2 + 2 x 3 = 10 x 2 + 3 x 3 = 6 x 1 - 3 x 2 - x 3 = 4 . (a) Determine the solution set of the system and write it in parametric form. The augmented matrix is B = 1 - 2 2 10 0 1 3 6 1 - 3 - 1 4 . The reduced echelon form of B is 1 0 8 22 0 1 3 6 0 0 0 0 . Thus, x 3 is free, x 1 = - 8 x 3 + 22, and x 2 = - 3 x 3 + 6. The parametric form of the solution set is x = t - 8 - 3 1 + 22 6 0 , t any scalar . (b) The coefficient matrix for the above system is A = 1 - 2 2 0 1 3 1 - 3 - 1 . Are the columns of A linearly independent or linearly dependent? Justify your answer. The columns of A are linearly dependent since the reduced echelon form of A has a zero row and therefore the homogeneous equation A x = 0 has nontrivial solutions. In fact, 8 1 0 1 + 3 - 2 1 - 3 - 2 3 - 1 = 0 0 0 is a typical dependence relation.

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2. (8 points) A linear transformation T : 2 3 satisfies T ( e 1 ) = (2 , 1 , 1) and
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