Hom(V,W) is a vector space

Question

WTS: Hom(V,W) which is the set of all linear maps is a vector space.

The question I have is that, isn't Hom(V,W) a subset of W? Because Hom(V,W) is just the set of all linear maps from V to W, so why can't I just show that it is a subspace?

Answer 1

You are probably thinking of the image of a particular function $V\to W$. By itself, a function $f\colon V\to W$ is a single object. You can define a vector space structure on the set of all such maps since it contains a zero element (the zero map) and you can scale any linear map by a constant: if $f\colon V\to W$ is linear and $c$ is a scalar, then $cf$ defined by $(cf)(v)=c(f(v))$ is another linear map.

Answer 2

Hom(V,W) is a set of homomorphisms(vector space homomorphism)/linear maps from V to W, which forms vector space! whilst what you're thinking is just image of V with particular element f of Hom(V,W)(i.e. f(V) is subspace of W). These two are different things!