Explain what is meant by a lattice in the Euclidean plane $\mathbb{R}^{2}$. Prove that such a lattice is either $\mathbb{Z} \boldsymbol{w}$ for some vector $\boldsymbol{w} \in \mathbb{R}^{2}$ or else $\mathbb{Z} \boldsymbol{w}_{1}+\mathbb{Z} \boldsymbol{w}_{2}$ for two linearly independent vectors $\boldsymbol{w}_{1}, \boldsymbol{w}_{2}$ in $\mathbb{R}^{2}$.