Let $M = (x_1 | \cdots | x_n) \in \mathrm{GL}_n(\mathbb{R})$. 2a. Show that $M \in \mathrm{GL}_n(\mathbb{Z})$ if and only if $M(\mathbb{Z}^n) = \mathbb{Z}^n$. 2b. Show the equivalence of the following propositions: i) $M \in \mathrm{GL}_n(\mathbb{Z})$. ii) The integer points of the parallelepiped $\mathcal{P} = \left\{\sum_{i=1}^n t_i x_i \mid \forall i \in \{1,\ldots,n\}, t_i \in [0,1]\right\}$ are exactly the $2^n$ points $\sum_{i=1}^n \varepsilon_i x_i$, where $\varepsilon_i \in \{0,1\}$ for all $i \in \{1,\ldots,n\}$.
Let $M = (x_1 | \cdots | x_n) \in \mathrm{GL}_n(\mathbb{R})$.
2a. Show that $M \in \mathrm{GL}_n(\mathbb{Z})$ if and only if $M(\mathbb{Z}^n) = \mathbb{Z}^n$.
2b. Show the equivalence of the following propositions:
i) $M \in \mathrm{GL}_n(\mathbb{Z})$.
ii) The integer points of the parallelepiped $\mathcal{P} = \left\{\sum_{i=1}^n t_i x_i \mid \forall i \in \{1,\ldots,n\}, t_i \in [0,1]\right\}$ are exactly the $2^n$ points $\sum_{i=1}^n \varepsilon_i x_i$, where $\varepsilon_i \in \{0,1\}$ for all $i \in \{1,\ldots,n\}$.