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Unformatted text preview: 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 ~u and ~v with respect to the basis. a) B = 1 1 2 1 , 2 1 1 1 ; ~u = 5 1 2 1 , ~v = 0 1 3 1 . Solution: Since neither vector in B is a scalar multiple of the other we get that B is linearly independent. Hence, it forms a basis for the subspace which it spans. The coordinates of ~u with respect to B and the coordinates of ~v with respect to B are determined by rowreducing the augmented systems c 1 1 1 2 1 + c 2 2 1 1 1 = 5 1 2 1 , d 1 1 1 2 1 + d 2 2 1 1 1 = 0 1 3 1 . We make one doubly augmented matrix and rowreduce to get 1 2 5 1 1 1 1 2 1 2 3 1 1 1 1 1 0 3 2 0 1 4 1 0 0 0 0 Thus [ ~u ] B = c 1 c 2 = 3 4 and [ ~v ] B = d 1 d 2 = 2 1 . b) C = { x 3 + x 2 1 , x 3 + x + 1 , x 2 + x + 2 } ; ~u = x 3 3 x 2 + 2 x + 3, ~v = x 3 + 3 x + 7. Solution: Consider x 3 + 0 x 2 + 0 x + 0 = c 1 ( x 3 + x 2 1) + c 2 ( x 3 + x + 1) + c 3 ( x 2 + x + 2) 0 = ( c 1 + c 2 ) x 3 + ( c 1 + c 3 ) x 2 + ( c 2 + c 3 ) x + ( c 1 + c 2 + 2 c 3 ) . We row reduce the coefficient matrix of the corresponding system to get 1 1 0 1 0 1 1 1 1 1 2 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 C is linearly independent and hence forms a basis for the subspace which it spans. The coordinates of ~u with respect to B and the coordinates of ~v with respect to B are determined by rowreducing the augmented systems c 1 ( x 3 + x 2 1) + c 2 ( x 3 + x + 1) + c 3 ( x 2 + x + 2) = x 3 3 x 2 + 2 x + 3 c 1 ( x 3 + x 2 1) + c 2 ( x 3 + x + 1) + c 3 ( x 2 + x + 2) = x 3 + 3 x...
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