assign7_soln - Math 136 Assignment 7 Solutions 1 Show the...

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Math 136 Assignment 7 Solutions 1. Show the each of the following sets form a basis for the subspace that they span, and determine the coordinates of ~x and ~ y with respect to the basis. a) { (1 , 1 , 0 , 1 , 0) , (1 , 0 , 2 , 1 , 1) , (0 , 0 , 1 , 1 , 3) } ; ~x = (2 , - 2 , 5 , - 1 , - 5), ~ y = ( - 1 , - 3 , 3 , - 2 , - 1). Consider 0 = c 1 (1 , 1 , 0 , 1 , 0) + c 2 (1 , 0 , 2 , 1 , 1) + c 3 ((0 , 0 , 1 , 1 , 3) = ( c 1 + c 2 , c 1 , 2 c 2 + c 3 , c 1 + c 2 + c 3 , c 2 + 3 c 3 ) We row-reduce the coefficient matrix to get 1 1 0 1 0 0 0 2 1 1 1 1 0 1 3 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 . Thus, the only solution is c 1 = c 2 = c 3 = 0 so the set of vectors is linearly independent and so the vectors in B form a basis for the subspace which they span. The coordinates of ~x with respect to B and the coordinates of ~ y with respect to B are determined by row-reducing the augmented systems c 1 (1 , 1 , 0 , 1 , 0) + c 2 (1 , 0 , 2 , 1 , 1) + c 3 (0 , 0 , 1 , 1 , 3) = (2 , - 2 , 5 , - 1 , - 5) d 1 (1 , 1 , 0 , 1 , 0) + d 2 (1 , 0 , 2 , 1 , 1) + d 3 (0 , 0 , 1 , 1 , 3) = ( - 1 , - 3 , 3 , - 2 , - 1) . We make one doubly augmented matrix and row-reduce to get 1 1 0 2 - 1 1 0 0 - 2 - 3 0 2 1 5 3 1 1 1 - 1 - 2 0 1 3 - 5 - 1 1 0 0 - 2 - 3 0 1 0 4 2 0 0 1 - 3 - 1 0 0 0 0 0 0 0 0 0 0 Thus [ ~x ] B = c 1 c 2 c 3 = - 2 4 - 3 and [ ~ y ] B = d 1 d 2 d 3 = - 3 2 - 1 . b) 1 1 1 0 , 0 1 1 1 , 2 0 0 - 1 ; ~x = 0 1 1 2 , ~ y = - 4 1 1 4 . Solution: Consider 0 0 0 0 = c 1 1 1 1 0 + c 2 0 1 1 1 + c 3 2 0 0 - 1 = c 1 + 2 c 3 c 1 + c 2 c 1 + c 2 c 2 - c 3 . 1
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2 We row-reduce the coefficient matrix to get 1 0 2 1 1 0 1 1 0 0 1 - 1 1 0 0 0 1 0 0 0 1 0 0 0 . Thus, the only solution is c 1 = c 2 = c 3 = 0 so the set of vectors is linearly independent (The vectors in B therefore form a basis for the subspace which they span).
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