lattice in $\mathbb{R}^{n}$

A lattice in ${\mathbb{R}}^{n}$ is an $n$ -dimensional additive free group over $\mathbb{Z}$ which generates ${\mathbb{R}}^{n}$ over $\mathbb{R}$ .

Example: The following is an example of a lattice $\mathcal{L}\subset{\mathbb{R}}^{2}$ , generated by $w_{1}=(1,2),w_{2}=(4,1)$ .

$\mathcal{L}=\{\alpha w_{1}+\beta w_{2}\mid\alpha,\beta\in\mathbb{Z}\}$

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