I am trying to prove that, for positive integers $m$ and $n$:
\begin{equation*} \text{Aut}(\mathbb{Z}_n^m) \cong \text{GL}_m(\mathbb{Z}_n) \end{equation*}
My attempt is to define function $\phi: \text{Aut}(\mathbb{Z}_n^m)\to \text{GL}_m(\mathbb{Z}_n)$ by letting $\phi(f)$ be the matrix with the $i$th column being $f(0, \cdots, 0,1,0,\cdots,0)$, where the $1$ is at the $i$th position. I was able to prove this map is a well defined injective homomorphism. However, I cannot prove this map is surjective. Any hint would be appreciated.