Two simplexes $\mathcal{S}$ and $\mathcal{S}'$ in $\mathbb{R}^n$ are called equivalent if there exist an enumeration of the vertices $s_0, s_1, \ldots, s_n$ of $\mathcal{S}$, and $s_0', s_1', \ldots, s_n'$ of $\mathcal{S}'$, and a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $A(s_i - s_0) = s_i' - s_0'$ for all $i = 1, \ldots, n$. Show that two integer simplexes $\mathcal{S}$ and $\mathcal{S}'$ are equivalent if and only if there exist a matrix $A \in \mathrm{GL}_n(\mathbb{Z})$ and a vector $b \in \mathbb{Z}^n$ such that $\mathcal{S}' = A(\mathcal{S}) - b$.
Two simplexes $\mathcal{S}$ and $\mathcal{S}'$ in $\mathbb{R}^n$ are called equivalent if there exist an enumeration of the vertices $s_0, s_1, \ldots, s_n$ of $\mathcal{S}$, and $s_0', s_1', \ldots, s_n'$ of $\mathcal{S}'$, and a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $A(s_i - s_0) = s_i' - s_0'$ for all $i = 1, \ldots, n$.
Show that two integer simplexes $\mathcal{S}$ and $\mathcal{S}'$ are equivalent if and only if there exist a matrix $A \in \mathrm{GL}_n(\mathbb{Z})$ and a vector $b \in \mathbb{Z}^n$ such that $\mathcal{S}' = A(\mathcal{S}) - b$.