assign7_soln

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) } ; = (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 1 + c 2 + c 3 2 +3 c 3 ) We row-reduce the coeFcient 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 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 53 1 1 1 - 1 - 2 0 1 3 - 5 - 1 1 0 0 - 2 - 3 0 1 0 42 0 0 1 - 3 - 1 0 0 0 00 0 0 0 Thus [ ] B = c 1 c 2 c 3 = - 2 4 - 3 and [ ± y ] B = d 1 d 2 d 3 = - 3 2 - 1 . b) ±² 11 10 ³ , ² 01 ³ , ² 20 0 - 1 ³´ ; = ² 12 ³ , ± y = ² - 41 14 ³ . Solution: Consider ² ³ = c 1 ² ³ + c 2 ² ³ + c 3 ² 0 - 1 ³ = ² c 1 +2 c 3 c 1 + c 2 c 1 + c 2 c 2 - c 3 ³ . 1
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2 We row-reduce the coefcient matrix to get 1 0 2 1 1 0 1 1 0 01 - 1 1 0 0 0 1 0 0 0 1 0 0 0 .
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assign7_soln - Math 136 Assignment 7 Solutions 1. Show the...

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