# Dual and simple modules

Let $A$ be a $k$ finite dimensional algebra and let $M$ be a simple finite dimensional right $A$-module. Why is the dual of $M$, i.e $\operatorname{Hom}_{k}(M,k)$ a simple left $A$-module?

-

Let $M$ be any finite dimensional $A$-module.
Suppose $N$ is a submodule of $\hom_k(M,k)$. Then $N^\perp=\{m\in M:\forall\phi\in N,\phi(m)=0\}$ is a submodule of $M$, and this establishes a bijection $$N\in\operatorname{Sub(hom_k(M,k))}\mapsto N^\perp\in\operatorname{Sub} M$$ between the set of submodules of $M$ and those of $\hom_k(M,k)$.
it is clearly surjective since giving a submodule of $M$ then take the set of all linear map which vanish on that submodule. Why is it injective though? – user10 Feb 4 '13 at 22:17