Let $\Gamma=\{mw_1+nw_2:m,n\in\mathbb{Z}\}$ and $\Gamma'=\{mw_1'+nw_2':m,n\in\mathbb{Z}\}$. Show that $\Gamma=\Gamma'$ if and only if there exists a matrix $A\in SL(2,\mathbb{Z})$ such that $\left( \begin{array}{c} w_1' \\ w_2' \\ \end{array} \right)=A\left( \begin{array}{c} w_1 \\ w_2 \\ \end{array} \right)$.
In the forward direction, I have reached the point where I have concluded that $\det(A)=\pm 1$. However, I have no way of concluding that $\det(A)=1$. Can someone help me out with this?
You can't get any further unless you require e.g. that the bases are positively oriented.