Intuition Behind Homomorphisms of Representations

Solution 1:

A representation of $G$ is just a vector space $V$ with an additional "scalar multiplication":

For $g \in G$ and $v \in V$, we have a new vector $gv \in V$. This operation has to satisfy the following axioms:

• For each $g \in G$, left-multiplication by $g$ is a linear map from $V$ to $V$, that is to say, $g(\lambda v_1+v_2) = \lambda(gv_1)+gv_2$ for any $v_1,v_2 \in V$ and any $\lambda$ in the underlying field.
• For any $g_1,g_2 \in G$ one has that $(g_1g_2)v = g_1(g_2v)$ for all $v \in V$.
• If $e$ is the identity element of $G$, then $ev = v$ for all $v \in V$.

Thus, a homomorphism between representations of $G$, $V_1$ and $V_2$, is just a linear map $T \colon V_1 \to V_2$ that also satisfies that $$T(gv) = gT(v)$$ for all $v \in V_1$ and all $g \in G$. Note that, in the above equality, yuxtaposition with $g$ has two different meanings, just as same as when one writes things like $T(\lambda v)=\lambda T(v)$ in the definition of linear map.