# Jacobson's Density Theorem for Semisimple Algebras

I am trying to follow the proof of the following:

Let $$A$$ be an algebra over an algebraically closed field $$k$$. Let $$V=V_1\oplus \ldots \oplus V_r$$, where $$V_i$$ are pairwise non-isomorphic irreducible finitely dimensional representations of $$A$$. Then the map $$\oplus_{i=1}^r\rho_i: A\to \oplus_{i=1}^r \operatorname{End}_k(V_i)$$ is surjective.

I am working out of Etingof's text (Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, and Elena Yudovina, Introduction to representation theory, 10 January 2011) (Theorem 2.5 page 24 in the link and if you have a hard copy of the text this is Theorem 3.2.2 on page 43). The proof is not too bad but there is one detail that I'm having difficulties with.

My issue is he quotes Proposition 2.2 (from the link above) to conclude that Im$$\oplus_{i=1}^r \rho_i=\oplus_{i=1}^r$$Im$$\rho_i$$. I am having difficulties seeing how this follows from the proposition. The rest of the proof and material leading up to this is fine. Any help would be greatly appreciated.

Thanks.
