# Thread: Change of bases of a lattice

1. ## Change of bases of a lattice

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.

2. Originally Posted by 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.

If you google "integer matrices" or "invertible int. matrices" you can find some interesting stuff, but

what you want is: an integer matrix is invertible iff its determinant is $\displaystyle \pm 1$ .

This is easy to prove and you're invited to try (hint: product rule for determinants).

Tonio

3. Makes sense. Thank you.