# Change of bases of a lattice

• Apr 3rd 2011, 01:27 PM
Pinkk
Change of bases of a lattice
Given a basis $(a,b)$ for a lattice $L$ in $\mathbb{R}^{2}$, show that every other basis for $L$ is of the form $(a', b') = (a,b)P$, where $P$ is a 2 by 2 integer matrix with determinant plus or minus 1.

So clearly $a' = ra + sb$ and $b' = ta + vb$ for $r,s,t,v \in \mathbb{Z}$, 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.
• Apr 4th 2011, 04:23 AM
tonio
Quote:

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

So clearly $a' = ra + sb$ and $b' = ta + vb$ for $r,s,t,v \in \mathbb{Z}$, 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 $\pm 1$ .

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

Tonio
• Apr 5th 2011, 06:38 PM
Pinkk
Makes sense. Thank you.