Consider the set of all invertible $2\times 2$ matrices over $\mathbb R$ (I think we can do it over $\Bbb C$ but I didn't look at it) ${\bf GL}(2,\Bbb R)=\left\{\left(\begin{matrix} a&b \\c &d\end{matrix}\right):a,b,c,d\in\Bbb R,ac-bd\neq 0 \right\}$
Consider now the set $\bf M$ of all functions of the form
$f(x)=\frac{ax+b}{cx+d}$
again with $a,b,c,d\in\Bbb R,ad-bc\neq 0 $. If we identify each of these with a matrix $\left(\begin{matrix} a&b \\c &d\end{matrix}\right)$ then we can define an isomoprhism between $\rm GL$ and $(\bf M,\circ)$ as groups with operations of matrix multiplications and functional composition, respectively, and identities$\left(\begin{matrix} 1&0 \\0 &1\end{matrix}\right)=e$ $x=\frac{1x+0}{0x+1}=e'$ since if $\eqalign{ & f = \frac{{ax + b}}{{cx + d}} \cr & g = \frac{{ex + f}}{{gx + h}} \cr} $ then $f \circ g = \frac{{\left( {ae + bg} \right)x + af + bh}}{{\left( {ce + dg} \right)x + dh + cf}}$ which corresponds to
$\left(\begin{matrix} a&b \\b &c\end{matrix}\right)\left(\begin{matrix} e&f \\g &h\end{matrix}\right)=\left(\begin{matrix} ae + bg&af + bh \\ce + dg &dh + cf\end{matrix}\right)$ and similarily for inversion, ${f^{ - 1}} = \frac{1}{{ac - bd}}\frac{{dx - b}}{{ - cx + a}}$
and $\left(\begin{matrix} \frac d\Delta& \frac {-b}\Delta \\\frac{-c}\Delta &\frac a\Delta \end{matrix}\right)=e$
I include the determinant $\Delta=ac-bd$ inside the matrix to avoid any multiplication by scalars considerations.
My question is: How can this be generalized to ${\bf GL}(n,K)$, and what is the theoretical relevance of this? I know Möbius Transformations are important in Complex Analysis, for instance, but I haven't seen any "higher dimesional" equivalent around.
ADD As users noted, for every $a\in \Bbb R$,$\left(\begin{matrix}a&0\\0&a\end{matrix}\right)\mapsto x$
so the isomoprhism is actually obtained by quoting ${\bf GL}(2,\Bbb R)$ by $I=\{aI_2:a\in\Bbb R\}$