# 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.

No. You must define a vector space structure on $$\hom(V,W)$$. It's easy though:

• $$(f+g)(v)=f(v)+g(v)$$;
• $$(\lambda f)(v)=\lambda.\bigl(f(v)\bigr)$$.

And now you must check that this is indeed a vector space structure.

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.

• so if hom(v,w) treated as a field in place of F? as f,g are elemenets of hom(v,w) not just elements of F – james black Sep 13 '19 at 4:04
• $\operatorname {hom}(V,W)$ is the vector space over a field $\Bbb F$. So the "points" of the vector space are homomorphisms. The scalars come from $\Bbb F$. – Chris Custer Sep 13 '19 at 4:11