In a vector space, linear transforms can act on points of the space by the usual matrix multiplication rule, but in this note I am reading they use a different action (The *Möbius transformation*). It's easy to multiply everything out and see that $A(Bx) = (AB)x$ holds but that doesn't explain why this works.

So my question is this, Is there a classification of all possible actions?

edit: just to clarify - I'm more interested in the general case than just the *Möbius* one but I do appreciate the answers so far!

**Edit II:** I think I have realized what's *really* happening here is that the matrices are just **representations** of, for example rationals, So they act on rationals in a way predetermined by what they represent.