Let $A$ be a central simple $k$-algebra of dimension $n^2$ over $k$. Also assume that $A= {\rm End} (V)$, where $V$ is an $n$-dimensional $k$-vector space. For $0\leq i \leq n$, let $M_i$ be the set of matrices with arbitrary entries on the $i^{\rm th}$ column and zero in other columns. These are the left simple $A$-modules, and they are isomorphic. Also assume that $B= {\rm End} (W)$, where $W$ is an $n$-dimensional $k$-vector space. Also suppose that we have given an algebra homomorphism $\varphi: A \rightarrow B$. I am stuck and I can not realize how should I construct an isomorphism $f : V \rightarrow W$.
What can we say about the reverse procedure? Given the isomorphism $f : V \rightarrow W$, how should I construct the algebra homomorphism $\varphi: A \rightarrow B$?