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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.

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The title says "possible linear actions" and the body of the question "possible actions". So what is space (Euclidean space, sphere, etc) and what kind of object acts on it (a linear transformation, diffeomorphism, topological group, etc)? – Victor Protsak Jun 14 '10 at 1:20
I fear this question doesn't satisfy either of the requirements of the first two paragraphs of the FAQ: it's neither at research level, nor well defined. – HJRW Aug 3 '10 at 16:09

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.

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A Möbius transformation (öbius_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.

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