Hello and thank you for your patience.
(DISCLAIMER: I'm a novice and not a mathematician by trade and I'm not certain how to articulate most of my questions here. I am learning from experiences and this is my second question here at SE.)
I was recently reading a finite mathematics article about visualizing the 48 actions of GL(2,3). What is the process that is being used here to generate these visualizations from the 2 x 2 matrices beneath each visualization?
Original article: http://finitegeometry.org/sc/9/3x3.html
The space consists of just nine points $(0,0)$, $(0,1)$, $(0,2)$, $(1,0)$, $(1,1)$, $(1,2)$, $(2,0)$, $(2,1)$, $(2,2)$ which can be naturally put on a $3\times3$ grid. The pictures show arrows which connect each point to the point it is mapped to (the latter being where the arrow points to), with the arrows for points mapping to itself omitted (thus the first picture, the identity, does not contain any arrows). One particular additional point to note is that the actually draw the grid points at coordinates $-1$, $0$, $1$, using that in $\mathbb F_3$, $-1\equiv 2$. For this reason, the point in the middle (corresponding to $(0,0)$ never moves.
The pictures might have been slightly more intuitive if they also had drawn the points themselves.