# Different approches to Geometry

1. Feb 15, 2012

### ShayanJ

I read somewhere that a Geometry is a non Empty set and a subset of its power set which has subsets with at least two elements.The elements of the first set are called points and the elements of the second set are called lines.With specifying these two sets and considering some axioms,you will get a geometry.Now I have two questions.
1-As with vector spaces(which you can define things as vectors too different from arrows in space),Can I build a geomery with e.g. the set of all 2x2 matrices?
2-What is the relationship of this approach to geometry with the manifold geometry?

thanks

2. Feb 17, 2012

Probably you would more likely get some answer if you could remember "where".

3. Feb 17, 2012

### micromass

That's the incidence geometry- approach to geometry.

Certainly, the set of all 2x2-matrices is a vector space, and each vector space induces a geometry. The points will be the vectors and the lines will be sets of the form $u+span(v)$ with v nonzero. Thus the lines through the origin will be the one-dimensional subspaces.

As a geometry, the set of all 2x2-matrices will be isomorphic to the geometry $\mathbb{R}^4$.

4. Feb 17, 2012

Hey Shyan.