Let $\mathbf x=\left( \begin{matrix}x_1 \\ x_2 \\ \end{matrix} \right)$ and $\mathbf y=\left( \begin{matrix}y_1 \\ y_2 \\ \end{matrix} \right)$ be two linearly independent vectors is $\mathbb R^2$ and let $K = \left[ \begin{matrix} x_1 & y_1 \\ x_2 & y_2 \end{matrix} \right] $. Let $W = \{a\mathbf x + b\mathbf y|a,b \in \mathbb Z \}$
(a) Show that $\mathbb R^2 / W$ is isomorphic $(\mathbb R/\mathbb Z) * (\mathbb R/\mathbb Z) $
(b) If $\mathbf u, \mathbf v$ are two other linearly independent vectors in $\mathbb R^2$, with lattice $A$ and matrix $B$, show that $W = A$ if and only if $B = GK$ for some $2 * 2$ matrix $G$ with integral entries, and its determinant is $1$ or $-1$.
I’m not sure how I can start with this question. I know that $\mathbb R/\mathbb Z$ is isomorphic to the circle group but will that be able to help me?
For (b), I’m not sure how the hint that the matrix has integral entries and has determinant 1 or -1 will come into effect.
Would appreciate some hints to help me complete this question!
For (a) look at it like this
$$ (a, b) \in \mathbb{R} \times \mathbb{R} \longleftrightarrow a{\bf x} + b{\bf y} $$ $$ (a, b) \in \mathbb{Z} \times \mathbb{Z} \longleftrightarrow a{\bf x} + b{\bf y} $$
For (b) if you have a second basis ${\bf x}', {\bf y}'$ for $W$ then what can you say about the transformation
$$ T : W \to W$$
defined by $T({\bf x}) = {\bf x'}$ and $T({\bf y}) = {\bf y}'$? What does the matrix form of $T$ look like? (You will need to write ${\bf x'} = a{\bf x} + b{\bf y}$ and ${\bf y}' = c{\bf x} + d{\bf y}$.) What about $T^{-1}$?