1. The problem statement, all variables and given/known data 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. 3. 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.