Homomorphism of linear representations

Definition

Abstract formulation

Suppose $G$ is a group, $V_{1},V_{2}$ are vector spaces over a field $k$, and $\rho _{1}:G\to GL(V_{1})$, $\rho _{2}:G\to GL(V_{2})$ are linear representations of $G$. A homomorphism of representations from $\rho _{1}$ to $\rho _{2}$ is a $k$-linear map $f:V_{1}\to V_{2}$ such that, for all $g\in G$:

$f\circ \rho _{1}(g)=\rho _{2}(g)\circ f$

Matrix formulation

Suppose $G$ is a group, $k$ is a field, and $\rho _{1}:G\to GL(m,k)$ and $\rho _{2}:G\to GL(n,k)$ are representations of $G$ over $k$. A homomorphism from $\rho _{1}$ to $\rho _{2}$ is a $n\times m$ matrix $F$ with the property that for all $g\in G$:

$F\rho _{1}(g)=\rho _{2}(g)F$

where the multiplication on both sides is matrix multiplication.