Given $S \in GL_n(\mathbb{R}^n)$.
Show that $x \mapsto Sx$ is an orientation preserving diffeomorphism on $\mathbb{T}^n$ if and only if $S \in SL_n(\mathbb{Z}^n)$.
I'm working on the only if part. Orientation perserving implies $\det(S) > 0$.
How can I prove that we must have $S \in SL_n(\mathbb{Z}^n)$. I can see that when $S \in SL_n(\mathbb{R}^n)$ is not enough to be surjective, but can't prove that $S \in SL_n(\mathbb{Z}^n)$.