Let be $H=Hom(\mathbb{K}^n,\mathbb{K}^m)$ the set of all linear maps from $\mathbb{K}^n$ to $\mathbb{K}^m$. I want to show that $H$ is ismorphic to the set of all matrixes $M_{n,m}(\mathbb{K})$. In order to do that, I need to stablish first the algebraic structure of both sets. I know that both sets are vector spaces ($M_{n,m}(\mathbb{K})$ with usual sum and scalar poduct and $H$ with usual sum and composition of maps). Is that the best we can say about both sets?. I read several times that $H$ is a ring (and so I could try to prove there exists a isomorphism of rings between them)... but one distributive law fails: $f\circ (h+g)\neq f\circ h+ f\circ g$, therefore $H$ can't be a ring.
Any answer to question in bold is appreciated. Thank you!