TITLE: How to imagine vector spaces (and projective spaces) over a finite field
QUESTION [11 upvotes]: So I have been learning about projective spaces for the last few hours, and I think I understand the basics pretty well, but there is an exercise, which I do not know how to solve at all. It comes down to being able to imagine and understand vector spaces and projective spaces over a finite field.
Let's say, my projective space is a set of all the one dimensional subspaces of my vector space. Then, each line in my projective space has at least one, most of the time multiple representants (in other words: direction vectors) that span the line. The picture below shows the projective plane over a finite field with 3 elements (the Galois-Field $GF(3)$). I would now like to find a description for the points in this pictures, whereas each point should have some corresponding representants. How do I determine which vector belongs to which point?

In $\mathbb{R}^3$ I can very well see that a vector $(1,0,0)$ spans my first axis, but what about a finite field? Any help is greatly appreciated!

REPLY [5 votes]: There are some decisions to be made when we think of writing representatives for points in a finite projective plane, because in general each point has more than one representing vector. For this reason, we usually represent the points with vectors that have been scaled to have a "1" as the first nonzero entry (or maybe as the last nonzero entry in some books). To make it clear that we mean this to represent a subspace, two common notations are to use angled brackets (to refer to the span of a vector) as in
$$\langle (1,0,0) \rangle$$
or to use what are known as homogeneous coordinates; these are usually written with colons between the entries such as
$$(1:0:0)$$
and the convention is that we ignore differences of a scalar multiple. Because there are so many symmetries of a finite projective plane, you can choose any three points (that are not all on the same line) to label $\langle (1,0,0) \rangle$, $\langle (0,1,0) \rangle$, $\langle (0,0,1) \rangle$.
Now when you want to fill in vector representatives for the other points, just work on filling them in. For example, the line containing $\langle (1,0,0) \rangle$ and $\langle (0,1,0) \rangle$ would be denoted $\langle (1,0,0), (0,1,0) \rangle$; it contains vectors represented by linear combinations of these two vectors, so we see that the other two points we need to fill in are $\langle (1,1,0) \rangle$ and $\langle (1,2,0) \rangle$. We can get unique representatives from the line $\langle \bf{u}, \bf{v} \rangle$ by considering the vectors $\{\bf{v}\} \cup \{\bf{u} + \lambda \bf{v} \ : \ \lambda \in \mathbb{F} \}$ (we would then scale each vector if necessary to make the first nonzero entry equal to 1).
For the most part you can choose which points to label with which coordinates in the beginning, as you work further in you may have to set up a system with a few linear equations and then solve it to make sure your labelings are consistent.