Projectivities $\pi:\mathbb KP^1\rightarrow\mathbb KP^1$ I am a little bit confused conerning the following example of projectivities $\pi:\mathbb KP^1\rightarrow\mathbb KP^1$.
On the affine part $\mathbb K\subseteq \mathbb KP^1$ they are exactly the rational maps of the form $x\rightarrow \frac{ax+b}{cx+d}, x\in\mathbb K$
My question is: How can I wirte down explicitly the six projectivities of the form above that permutes the points $0,1,\infty$  ?

I dislike the formulation as a fraction, as it doesn't handle $\infty$ too well. I prefer homogenous coordinates and a matrix notation. Doesn't change the core of the issue, though.
\begin{align*}
0&\mapsto0 & 1&\mapsto1 & \infty&\mapsto\infty &
\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix} &&
x&\mapsto\frac{x}{1} \\
0&\mapsto1 & 1&\mapsto0 & \infty&\mapsto\infty &
\begin{pmatrix}-1 & 1 \\ 0 & 1\end{pmatrix} &&
x&\mapsto\frac{-x+1}{1} \\
0&\mapsto\infty & 1&\mapsto1 & \infty&\mapsto0 &
\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix} &&
x&\mapsto\frac{1}{x} \\
0&\mapsto\infty & 1&\mapsto0 & \infty&\mapsto1 &
\begin{pmatrix}1 & -1 \\ 1 & 0\end{pmatrix} &&
x&\mapsto\frac{x-1}{x} \\
0&\mapsto0 & 1&\mapsto\infty & \infty&\mapsto1 &
\begin{pmatrix}1 & 0 \\ 1 & -1\end{pmatrix} &&
x&\mapsto\frac{x}{x-1} \\
0&\mapsto1 & 1&\mapsto\infty & \infty&\mapsto0 &
\begin{pmatrix}0 & 1 \\ -1 & 1\end{pmatrix} &&
x&\mapsto\frac{1}{-x+1} \\
\end{align*}
