Prove isomorphism between invertible linear transformations and invertible matrices Let G = {invertible linear transformations from $\Bbb Z_{p}^d$ to $\Bbb Z_{p}^d$}
How can I come up with a concrete example of isomorphism from G to $GL(d,p)$, i.e. show that it is a bijective homomorphism?

The fact that you're working on the field $\mathbb Z_p$ is not so important. For a finite dimensional vector space, you can define the isomorphic $\varphi$ you're looking for by taking the canonical basis $(e_1, \dots, e_d)$ and for $u \in G$ the matrix $M=\varphi(u)$ is the one having for coefficients $M_{i,j}$ where 
$$u(e_j)=\sum_{i=1}^d M_{i,j}e_i$$
