Is there a classification of possible linear actions?

Question

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.

Answer 1

Not an exact answer to your question, but Möbius transformations on $\mathbb{C}$ can be viewed as coming from linear transformations (in the usual sense) on $\mathbb{C}^2$: Identify $z\in\mathbb{C}$ with the one-dimensional subspace of $\mathbb{C}^2$ spanned by the vector $(z,1)$. Invertible linear maps send one-dimensional subspaces to other one-dimensional subspaces, and thus you reproduce the Möbius transformations. The “point at infinity” corresponds to the line $\mathbb{C}\oplus 0$ under this correspondence. The set of all one-dimensional subspaces is the one-dimensional complex projective space $\mathbb{PC}^1$, also known as the Riemann sphere.

Answer 2

A Möbius transformation (http://en.wikipedia.org/wiki/Möbius_transformation) is usually considered as acting on the projective line, which is associated to a two-dimensional vector space. Roughly speaking, a matrix action that is quite conventional on the two-dimensional vector space becomes linear fractional when you look at it in terms of lines through 0. That's why "it works". A general classification is probably too broad an idea to be useful here.