Let $A$ be an associative algebra over a field $K$ and let $\rho:A \to \operatorname{End}_K(V)$ be a representation of $A$. Is it true that if $\rho$ is faithful then it's also an isomorphism?
Note: I know it's ture if $A$ is a finite group algebra $K[G]$ which relies on the first Isomorphism theorem. But How about an arbitrary algebra? Even is it true for finite diemensional algebra?