- #1
negation
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Homework Statement
In R4,
let u = (-1,1,5,-3) and v = (2,-3,-5,-2)
and
let a = (9,-12,-30,3) and b = (2,-1,-14,11)
For each of the vectors a, b you are asked to determine whether it belongs to the subspace of R4 spanned by u, v.
The Attempt at a Solution
Since R4 spans u and v, then, R4 = span(u,v)
This implies also that if a is a spanning set,then,
span(a) = span(u,v)
(x,y,z,u) = λ1u + λ2v =
-λ1 + 2λ2 = x
λ1 -3λ2 = y
5λ1 - 5λ2 = u
-3λ1 -2λ2 = z
(x,y,z,u) = (-λ1 + 2λ2,λ1 -3λ2 ,5λ1 - 5λ2 ,-3λ1 -2λ2 )
(-λ1 + 2λ2,λ1 -3λ2 ,5λ1 - 5λ2 ,-3λ1 -2λ2 ) = γ.a
(-λ1 + 2λ2,λ1 -3λ2 ,5λ1 - 5λ2 ,-3λ1 -2λ2 ) = γ(9,-12,-30,3) = 9γ1 - 12γ2-30γ3+3γ4
This looks very chaotic. Am I on the right track?
EDIT: I think I might have found a much easier way.
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