Question 1042146: Let T={(2,3,1),(-1,2,-1),(0,7,-1)} be a set of vectors in R3 and S=span(T)
Determine S and give a basis for S which only contains elements in T.
Thank you!
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! S = a(2,3,1) + b(-1,2,-1) + c(0,7,-1) where a, b, c are real numbers.
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S = (2a,3a,a) + (-b,2b,-b) + (0,7c,-c)
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S = (2a-b, 3a+2b+7c, a-b-c)
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now we want to reduce the span to get our basis
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let
(2a-b, 3a+2b+7c, a-b-c) = (0,0,0)
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then 2a-b = 0 and b = 2a
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substitute for b
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(0, 7a+7c, -a-c) = (0,0,0)
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this holds when c = -a
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therefore,
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a(2,3,1) + 2a(-1,2,-1) -a(0,7,-1) for any a
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if we set a = 1, we see that
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(0,7,-1) = (2,3,1) +2(-1,2,-1)
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we do not need (0,7,-1) since it is a linear combination of the other two
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therefore every vector in T can be written in the form
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u(2,3,1) + v(-1,2,-1) = (2u-v, 3u+2v, u-v) for real u,v
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all we need to do is show (2,3,1) and (-1,2,-1) are linearly independent
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we do this by setting
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(2u-v, 3u+2v, u-v) = (0,0,0) implies that u=v=0
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note that u-v = 0 means that u=v =0
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