Change of bases of a lattice

**Pinkk** · April 3rd 2011, 01:27 PM

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.

**tonio** · April 4th 2011, 04:23 AM

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

**Pinkk** · April 5th 2011, 06:38 PM

Makes sense. Thank you.

Thread: Change of bases of a lattice

LinkBack

Thread Tools

Display

Change of bases of a lattice

Similar Math Help Forum Discussions

proof lattice

What's the determinant of this lattice?

Lattice

Lattice Points

Lattice Points

Search Tags