I want to prove that all translations on an affine straight line are induced by projective transformations of the projective extension.
If we look at $\mathbb{R}$ as our affine straight line, then the extension is $\mathbb{P}^1(\mathbb{R})=\mathbb{P}(\mathbb{R}^2)$. Furthermore, I know $\mathbb{R}\subset\mathbb{P}(\mathbb{R}^2)$.
So a projective transformation $\mathbb{P}(\mathbb{R}^2)\rightarrow\mathbb{P}(\mathbb{R}^2)$ restricted to $\mathbb{R}$ should be a translation $f:\mathbb{R}\rightarrow\mathbb{R}$. But all projective transformations are induced by an injective linear map $\mathbb{R}\rightarrow\mathbb{R}^2$.
So, what I need to prove now is that for all translations on $\mathbb{R}$ there is a $2\times2$-matrix that induces this. How do I do that?
(In other words: how do these matrices look?)
Yes, you can use homogeneous coordinates to represent all translations by $\tau$ on $\mathbb{R}$ as $$ T = \begin{pmatrix} 1 & \tau \\ 0 & 1 \end{pmatrix} $$ Your vectors use homogeneous coordinates as well $$ \begin{pmatrix} x \\ 1 \end{pmatrix} $$ It then works like $$ \begin{pmatrix} x' \\ 1 \end{pmatrix} = \begin{pmatrix} 1 & \tau \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ 1 \end{pmatrix} = \begin{pmatrix} x + \tau \\ 1 \end{pmatrix} $$