I'm taking intro to coding theory and am having some trouble understanding the basics of Projective Geometry, since our text does not give it much discussion. Namely, if PG(r-1,q) is the set of all subspaces of V(r,q), then how to I denote the lines in PG(3,q)?
I understand that PG(2,q) has lines that can be "cut-out" by single linear equations such as x1=0 for homogeneous coordinates (x1:x2:x3) (this is how prof described it in class). In fact, the coefficients of such equations can be represented as the points in PG(2,q).
BUT for PG(3,q) one equation leaves us with a plane. So how do I go about making a line in PG(3,q)? Using two linear equations? This makes two planes, but how I know that they intersect and aren't parallel? How can I generate all of them? Ex: How to draw all lines in PG(3,2)?
Unfortunately, I can't find a simple online resource on this anywhere so a direct explanation would be very helpful. Especially if it used the above terms.
Thanks! Mike
A line of $\mathrm{PG}(3,q)$ can be represented as a span of two points, or as the nullspace of a rank 2 $2\times 4$ matrix. This is the same as using two linear equations; each gives a plane, and the planes of $\mathrm{PG}(3,q)$ either coincide or intersect in a line. The planes will coincide precisely when the two equations are multiples of each other.
In $\mathrm{PG}(r-1,q)$, you can still represent a line as a span of two points. If you want to use the nullspace of a matrix, you will need to use a full rank $(r-2)\times r$ matrix (this represents a line as an intersection of $r-2$ hyperplanes).
If you want all the lines of $\mathrm{PG}(3,q)$ (each represented exactly once), you need to consider a set of $2 \times 4$ matrices having rank 2, no pair being row-equivalent. To do this, you can consider that matrices are row-equivalent if and only if they have the same reduced echelon form.