$P$ is a principal $G$-bundle, $ρ:G→GL(V)$ is a representation, and $E$ is the associated vector bundle of $P$, so $E=(P×V)/G$ with the right action $(p,v)⋅g=(p⋅g,ρ(g^{−1})v)$.
I have read that there is an isomorphism between the basic $V$-valued forms on $P$, and $E$-valued forms on $M$:
\begin{equation} Ω^k_{\text{bas}}(P,V)\stackrel\sim\to Ω^k(M,E) \end{equation} I don't see how they are isomorphic. What is the explicit isomorphism, and why is it an isomorphism?