For a homework problem I need to find all $A\in GL_2(\mathbb{R})$ that stabilizes some lattice $L$ in $\mathbb{R}^2$.
I have deduced that $L$ is a free $\mathbb{Z}$-module, and so left multiplication by $A$ is an isomorphism between modules. However, I am not quite sure how to approach from here on. Anyone with some ideas/hints, preferably starting from this step?