Before we try to understand transformations of RP², we need a better way to visualize the points and lines in RP².
We've gotten quite familiar with projectivities, the transformations of the extended Euclidean plane. What would the analogous transformations of RP² look like?
The projective linear group PGL(3,R) consists of transformations of RP². Can we somehow characterize these transformations?
Okay, so the projective linear group PGL(3,R) is really an analytic formulation of the set of projectivities. Let's try visualizing the action of PGL(3,R) on the affine plane z=1, and in the process gain more clarity about the types of projectivities that exist.
The Fundamental Theorem of PGL(3,R) remains to be proven. For this we'll need some linear algebra.