Homomorphism of linear representations

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.