In class we saw A 1-1 homomorphism from $\operatorname{Iso}(\mathbb{R})$ to $GL(2,\mathbb{R})$
$\operatorname{Iso}(\mathbb{R})\cong \left\{ \begin{pmatrix}\pm1 & x\\ 0 & 1 \end{pmatrix}|x\in\mathbb{R}\right\}. $
How can I get this result ? (It works, but it's probably not a guess, whats the idea ?)
Seeing this I think that there is a 1-1 homomorphism from $\operatorname{Iso}(\mathbb{R}^2)$ to $GL(3,\mathbb{R})$ , how can we find it ? (that's why I'm trying to gain a better understanding of $\operatorname{Iso}(\mathbb{R})$)
I could use some help with this.