## Group theory - orbits and fixed points (please help!)

I have my Group Theory exam in less than a month and I can't get to grips with orbits and stabilizers. Please could someone talk me through the solution to this question?

Let $\displaystyle G=GL(2,\mathbb R)$ and $\displaystyle X=\mathbb R^{2}$.

Let $\displaystyle G\times X\rightarrow X,$$\displaystyle \left(\begin{array}{ccc} \left(\begin{array}{cc} a & b\\ c & d\end{array}\right) & , & \left(\begin{array}{c} x\\ y\end{array}\right)\end{array}\right)\mapsto\left( \begin{array}{ccc} ax & + & by\\ cx & + & dy\end{array}\right),$ define a G action.

What are the orbits and fixed point sets of this G action?