### Linear Algebraic Groups (Part 2)

#### Posted by John Baez

This time we show how projective geometry ‘subsumes’ Euclidean, elliptic and hyperbolic geometry. It does so in two ways: the projective plane includes all 3 other planes, and its symmetry group contains their symmetry groups.

By the time we understand this, we’re almost ready to think about geometry as a subject that depends on a choice of group. But we’re also getting ready to think about algebraic geometry (for example, projective varieties).

- Lecture 2 (Sept. 27) - The road to projective geometry. Treating Euclidean, elliptic and hyperbolic geometry on an equal footing: in each case the symmetry group is a linear algebraic group of 3 × 3 matrices over a field $k$, points are certain 1d subspaces of $k^3$, and lines are certain 2d subspaces of $k^3$. In projective geometry we take the symmetry group to be
*all*of $\mathrm{GL}(3)$, take points to be*all*1d subspaces of $k^3$, and take lines to be*all*2d subspaces of $k^3$. It thus subsumes Euclidean, elliptic and hyperbolic geometry. In general we define projective $n$-space, $k\mathrm{P}^n$, to be the set of 1d subspaces of $k^{n+1}$.