Does u belong in span(x, v, w) given x not in span(v, w)?

[misterau]

Homework Statement

Suppose that u, v, w and x are vectors in a vector space V . If x !∈ span(v,w) and x ∈ span(u, v,w),
does it follow that u ∈ span(x, v,w)? Justify your answer.

Homework Equations

The Attempt at a Solution

Not really sure how to even start this problem. Any hints would be good thanks!

 

[lanedance]

Hi misterau

I would start with the definition of a span, or how any vector in a given span can be written...

 

[misterau]

I really not sure, probably not correct...
x != c1*v + c2*w
x = c1*v + c2*w + c3*u
u = c1*v + c2*w + c4*x

c1*v + c2*w != c1*v + c2*w + c3*u
0 != c3*(c1*v + c2*w + c4*x)
0 != c1*v + c2*w + c4*x

 

[xepma]

First hint:
Given:
x != c1*v + c2*w
x = c1*v + c2*w + c3*u

you know that c3 != 0.

Second hint:
Starting from this, can you find a more explicit expression for u in terms of x,v,w and c1,c2 and c3?