A matrix representation of a group $G$ is a homomorphism $$\rho : G \rightarrow GL_{n}(\mathbb{C})$$
A G-space is a vector space $V$ equipped with an action of $G$ $$ G \times V \rightarrow V, (g,v) \rightarrow gv$$ such that the map is linear, $ev = v$, where $e$ is the identity in $G$ and $(g_1g_2)(v) = g_1(g_2v)$.
A representation of a group $G$ is a vector space $V$ and a homomorphism $$\rho : G \rightarrow GL(V)$$
Now, apparently, all of these are equivalent definitions, but I'm having a hard time understanding how. Is the $\rho$ in the third definition the same as the action of $G$ in the second?
Is the $\rho$ in the first definition the matrix representation of the $\rho$ in the third definition? (since every linear map has an associated matrix).
Following on from these, I'm also confused about $G-$space homomorphisms and $G-$linear maps.
A $\textbf{homomorphism of G-spaces}$ $ f : V \rightarrow W$ is a linear map such that $f(gv) = gf(v), g \in G, v \in V$.
Let $\rho_1 : G \rightarrow GL(V), \rho_2 : G \rightarrow GL(W)$ be two representations. Then a $\textbf{G-linear map}$ between $\rho_1$ and $\rho_2$ is a linear map $f : V \rightarrow W$ such that $f\circ(\rho_1(g)) = \rho_2(g)\circ(f)$
I don't understand how these are equivalent either? Thank you for any help.
Sketch:
"$2\Rightarrow 3$" Let $g\cdot v$ be a group action on $V$. It induces a group homomorphism $\rho: G\to GL(V)$. For $g\in G$ the linear map $\rho(g):V\to V$ is given by $\rho(g)(v)=g\cdot v$.
"$3\Rightarrow 2$" For a given $\rho: G\to GL(V)$ homomorphism we define the group action $(g,v)\mapsto \rho(g)(v)$ on $V$. Remember that $\rho(g)$ is a linear map $V\to V$ so $\rho(g)(v)$ makes sense.
Of course you have to fill in all the details, namely that (a) these constructions are correct, (b) $G$-homomorphism are equivalent and that (c) the composition of both constructions yields identity. Or in other words that this construction is functorial and both functors are inverse to each other.
Finaly (1) is equivalent to (3) because $GL_n(k)\simeq GL(k^n)$ via the matrix representation of a linear map. And every finite dimensional vector space is isomorphic to $k^n$ for some $n$. Note that this makes sense only in finite dimensional case.