$f:\Bbb Z^2\to\Bbb Z^2:{n\brack m}\mapsto\begin{bmatrix}3&2\\4&3\end{bmatrix}{n\brack m}$
it is onto because:
$\left\{\begin{align*}&m = 9d - 4s\\ &n = 6d + 3s\end{align*}\right.$ (after solving the system of equations)
onto because m and n are integers
one-to-one because:
$\left\{\begin{align*}&3n + 2m = 3p + 2q\\ &4n + 3m = 4p + 3q\end{align*}\right.$
which is equal to $L_2 - L_1$ and $L_1 - 2L_2$:
$\left\{\begin{align*}&n = p\\ &n + m = p + q\end{align*}\right.$
equal to:
$\left\{\begin{align*}&n = p\\ &m = q\end{align*}\right.$
I have to solve the system of equations for both (to prove it is onto and one-to-one) if I understood correctly?