Define a lattice in $\mathbb{R}^{2}$ and the rank of such a lattice.

Let $\Lambda$ be a rank 2 lattice in $\mathbb{R}^{2}$. Choose a vector $\boldsymbol{w}_{1} \in \Lambda \backslash\{\boldsymbol{0}\}$ with $\left\|\boldsymbol{w}_{1}\right\|$ as small as possible. Then choose $\boldsymbol{w}_{2} \in \Lambda \backslash \mathbb{Z} \boldsymbol{w}_{1}$ with $\left\|\boldsymbol{w}_{2}\right\|$ as small as possible. Show that $\Lambda=\mathbb{Z} \boldsymbol{w}_{1}+\mathbb{Z} \boldsymbol{w}_{2}$.

Suppose that $\boldsymbol{w}_{1}$ is the unit vector $\left(\begin{array}{l}1 \\ 0\end{array}\right)$. Draw the region of possible values for $\boldsymbol{w}_{2}$. Suppose that $\Lambda$ also equals $\mathbb{Z} \boldsymbol{v}_{1}+\mathbb{Z} \boldsymbol{v}_{2}$. Prove that

$$\boldsymbol{v}_{1}=a \boldsymbol{w}_{1}+b \boldsymbol{w}_{2} \quad \text { and } \quad \boldsymbol{v}_{2}=c \boldsymbol{w}_{1}+d \boldsymbol{w}_{2}$$

for some integers $a, b, c, d$ with $a d-b c=\pm 1$.