Definition:Group Action of GL(2,Q) on Irrational Numbers

Definition
Let $\operatorname{GL}(2, \Q)$ be the general linear group on the field of rational numbers.

Let $\R\setminus \Q$ be the set of irrational numbers.

The standard group action of $\operatorname{GL}(2, \Q)$ on $\R\setminus \Q$ is the group action:
 * $\operatorname{GL}(2, \Q) \times (\R \setminus \Q) \to \R \setminus Q$:
 * $ \left( \begin{pmatrix}a&b \\

c&d\end{pmatrix}, x \right) \mapsto \dfrac{ax+b}{cx+d}$

Also see

 * Group Action of GL(2,Q) on Irrational Numbers is Group Action
 * Definition:Group Action of General Linear Group on Projective Line