Set of Vector Space Homomorphism Definition V, W are vector spaces over F and we have hom (V,W) as the set of homomorphisms of linear spaces.   Define addition  and scalar multiplication on  Hom(V, W) so it's vector space over F. 
I assume i would define it the same way as a vector space homomorphism which is:
Let V and W be two vector spaces over F. 
f :
V → W is a linear map if for every x, y ∈ V and c ∈ F, we have f(x + y) = f(x) + f(y) (i.e. f is a group homomorphism) and f(cx) = cf(x).
So x, y ∈ Hom(V,W), and c ∈ F, we have f(x + y) = f(x) + f(y) (i.e. f is a group homomorphism) and f(cx) = cf(x).
How would i prove my answer? Do i go through the 6 axioms of vectors one by one? Thanks.

You need to define addition of two elements of $\operatorname {Hom}(V,W)$, and also scalar multiplication. 
So given $f,g\in \operatorname {Hom}(V,W)$, define $(f+g)(v):=f(v)+g(v)$.  
Then define $(c\cdot f)(v):=c\cdot f(v)$.
The axioms should go through rather easily.
