Isomorphism between $R$ and its dual space

Question

Let $R$ be a finite dimensional algebra over a field $K$. If $f$ is an $R$-module monomorphism from $R$ to the dual $K$-space $\operatorname{Hom}_K(R,K)$ why it is onto? Thanks!

Answer 1

Let $f:R\to\operatorname{Hom}_K(R,K)$ be an $R$-module monomorphism. This is also a $K$-vector space monomorphism. Since $R$ is a finitely dimensional $K$-vector space, $\dim_KR=\dim_KR^*$, and thus every monomorphism of $K$-vector spaces $R\to R^*$ must be an isomorphism.