You are given three vectors in R3\mathbb{R}^3R3:
u=(1, 2, −1),v=(3, −1, 2),w=(11, 1, 4)\mathbf{u} = (1,\ 2,\ -1), \qquad \mathbf{v} = (3,\ -1,\ 2), \qquad \mathbf{w} = (11,\ 1,\ 4)u=(1, 2, −1),v=(3, −1, 2),w=(11, 1, 4)
Find scalars aaa and bbb with
w=au+bv\mathbf{w} = a\mathbf{u} + b\mathbf{v}w=au+bv
or decide that no such scalars exist. All three coordinates must match — a pair that works for two of them is not an answer.
Select all that apply.
a=3, b=2a = 3,\ b = 2a=3, b=2
a=2, b=−3a = 2,\ b = -3a=2, b=−3
No such aaa and bbb exist — w\mathbf{w}w lies outside span{u,v}\operatorname{span}\{\mathbf{u}, \mathbf{v}\}span{u,v}
a=2, b=3a = 2,\ b = 3a=2, b=3