It is well know that the Fundamental Group of the Klein Bottle is defined by
$$G=BS(1,-1)=\langle a,b:bab^{-1}=a^{-1}\rangle$$
An explicit description can be obtaned by define $G$ as the group of homeomorphisms of $\mathbb{R}^{2}$ generate by the functions $f,g: \mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$
$$f(x,y)=(x,y+1), g(x,y)=(x+1,1-y)$$
A linear action of $G$ over $\mathbb{R}^{2}$ is an aplication $"\cdot":G\times \mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ such that:
1.- $a\cdot(x_{1}+x_{2})=a\cdot x_{1}+a\cdot x_{2}$
2.- $a\cdot (b\cdot x)=(ab)\cdot x$
3.- $1_{G}\cdot x=x$
for all $a,b\in G$ and $x,x_{1},x_{2}\in \mathbb{R}^{2}$
My question is : how to define a linear action of $G$ over $\mathbb{R}^{2}$?