Irreducible representation over splitting field surjects to matrix ring

Statement
Suppose $$G$$ is a finite group and $$k$$ is a splitting field for $$G$$. Suppose $$\rho:G \to GL(n,k)$$ is an irreducible representation for $$G$$ over $$k$$. Then, the map extends uniquely by $$k$$-linearity to a $$k$$-linear map from the group ring to the matrix ring:

$$\tilde{\rho}: kG \to M(n,k)$$

The claim is that $$\tilde{\rho}$$ is surjective.

Instead of requiring $$k$$ to be a splitting field, we can require only that $$k$$ have characteristic not dividing the order of $$G$$ and the representation $$\rho$$ be absolutely irreducible.

Related facts

 * Central implies image under every irreducible representation is scalar
 * Schur's lemma