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**Pinkk** Given a basis $\displaystyle (a,b)$ for a lattice $\displaystyle L$ in $\displaystyle \mathbb{R}^{2}$, show that every other basis for $\displaystyle L$ is of the form $\displaystyle (a', b') = (a,b)P$, where $\displaystyle P$ is a 2 by 2 integer matrix with determinant plus or minus 1.

So clearly $\displaystyle a' = ra + sb$ and $\displaystyle b' = ta + vb$ for $\displaystyle r,s,t,v \in \mathbb{Z}$, but I don't know how to show that the matrix consisting of those integers has determinant plus or minus 1. Any help would be appreciated.