Seemingly counterintuitive Linear Algebra proof: Is it correct?

[diligence]

I'm taking a course based on Serge Lang's Linear Algebra. This isn't an assigned problem but I'm brushing up for an exam next week. Can you tell me if my proof is correct, because the result seems quite counter-intuitive?... (Actually, I assume the result is correct whether or not my proof is correct since it's an exercise in Lang).

Homework Statement

Let V be a vector space and let F be a linear map from V to the real numbers. Let W be the subset of V consisting of all elements v such that F(v)=0. Assume that W is not equal to V, and let v_o be an element of V which does not lie in W. Show that every element of V can be written as a sum: w + c(v_o) with some w in W and some number c.

Homework Equations

F(w)=0.
F(v_o) /=0.

The Attempt at a Solution

Let v be an element of V and let v_o be an element of W. Let c = F(v)/F(v_o) which is okay since F(v_o) is not equal to zero. Then we have 0 = F(v) - cF(v_o) = F(v - cv_o), since F is linear. Thus, v - c(v_o) is an element of W. Hence, there exists a w in W such that
v = w + c(v_o).

I guess the main thing I'm asking is if I'm allowed to use linearity of F to go backwards? I'm basically 95% sure this is okay, just want to make sure...thanks for looking..

 

[diligence]

wait that's actually not remarkable or counter-intuitive because you can just take the negative...hahah duh. i always want to make this stuff more complicated than it needs to be for some reason!