Equivalence of definitions of projective representations

Question

There are two definitions of projective representations, which I fail to prove equivalence between:

1. Let $G$ be a group and $\alpha:G \times G \to \mathbb C^{*}$ be a map, which satisfies \begin{equation} \alpha(g_2,g_3) \cdot \alpha(g_1,g_2g_3)=\alpha(g_1,g_2) \cdot \alpha(g_1g_2,g_3) \; \; \forall g_1,g_2,g_3 \in G. \end{equation} Then a projective representation of $G$ is a map $T:G \to GL(V)$, such that $T(g) T(h)=\alpha(g,h) T(gh)$. Note that this map need not be a homomorphism.
2. Let $G$ be a group and denote $PGL(V):=GL(V)/(\mathbb C^{*} \cdot \text{id}_{V})$. Then a projective representation of $G$ is a group homomorphism $\theta:G \to PGL(V)$.

How to show that $2 \implies 1$?

Answer 1

(1)$\Rightarrow$(2): Compose $G\xrightarrow{T}\mathrm{GL}(V)\to\mathrm{PGL}(V)$ to get $\overline{T}:G\to\mathrm{PGL}(V)$.

(2)$\Rightarrow$(1): Pointwise pick representatives $T(g)\in\mathrm{GL}(V)$ for $\overline{T}(g)\in\mathrm{PGL}(V)$, arbitrarily. Solve for $\alpha(g,h)$ in the cocycle condition to find how to define $\alpha(g,h)$ in terms of $T$.