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asmt1 - McGill University Math 270-2010 Fall Applied Linear...

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McGill University Math 270-2010 Fall: Applied Linear Algebra Written Assignment 1: due on Monday, Oct. 4, 2010 1. Give (3 × 5) matrix: A = 1 - 2 0 1 - 1 2 - 4 1 0 - 1 3 - 6 1 1 0 . (a) Determine the dimension of the subspaces col ( A ) and row ( A ); (b) Find the basis for col ( A ) and the basis of row ( A ); (c) Find the nullspace of A and the nullity null ( A ). (d) Let b = (1 , 1 , a ) T . Discuss the existence and uniqueness of the solutions for the system A x = b . Determine the rank of its augmented matrix A # and demonstrate that for what numbers a , the system (i) has unique solution exists; (ii) does not have solution; or (iii) has no- unique solution. In this case, give a set of linear independent solutions. 2. Calculate the following matrix multiplications: (1 , 2 , 3)(2 , 3 , 4) T =?; (1 , 2 , 3) T (2 , 3 , 4) =? 0 , 0 , 1 , 0 0 , 1 , 0 , 0 1 , 0 , 0 , 0 0 , 0 , 0 , 1 1 , 2 , 3 , 4 2 , 3 , 4 , 5 3 , 4 , 5 , 6 4 , 5 , 6 , 7 ; 1 , 0 , 0 , 0 0 , 1 , 0 , 0 2 , 0 , 1 , 0 0 , 0 , 0 , 1 1 , 2 , 3 , 4 2 , 3 , 4 , 5 3 , 4 , 5 , 6 4 , 5 , 6 , 7 ; 3. Given the matrix A = 1 - 2 0 2 - 4 1 3 - 6 1 . Find the elementary matrices E i such that (a) E 1 A equals to the elementary row operation: R 1 ⇐⇒ R 2 ; (b) E 2 A equals to the elementary row operation:
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