Let $\{ \mathbf{u}_{1},\dots,\mathbf{u}_{k} \}$ be linearly independent vectors in $\mathbb{R}^{k}$. A $k$-dimensional lattice with basis $\{ \mathbf{u}_{1},\dots,\mathbf{u}_{k} \}$ is the infinite discrete set $$\Lambda=\left\{ \mathbf{y}\in\mathbb{R}^{k}:\mathbf{y}=\sum_{i=1}^{k}n_{i}\,\mathbf{u}_{i},\ n_{1},\dots,n_{k}\in\mathbb{Z} \right\}$$i.e. $\Lambda$ is the collection of integer-coefficient linear combinations of the basis vectors.