I am reading Papantonopoulou: Algebra Ch 14 Symmetries. I am seeking to fully understand Theorem 14.21 (see attached Papantonopoulou pp 462 -463)

On page 462, Papantonopoulou defines translations, rotations and reflections for R2 and R3 (see attached). Note that the rotations are defined as about the origin and the reflections are about the X-axis or $\displaystyle e_1 $ axis.

Then on Page 463 he states Theorem 14.21 as follows:

14.21 TheoremAn isometry S of $\displaystyle \mathbb{R}^2$ or $\displaystyle \mathbb{R}^3$ can be uniquely expressed as $\displaystyle S = t_b \circ \rho_{\theta} \circ r^i $ where i = 0 or 1

I would like to use Theorem 14.21 to specify S for the isometry of $\displaystyle \mathbb{R}^2$ that maps the line y = x to the line y = 1 - 2x? [ ie what is $\displaystyle r^i$ , $\displaystyle \rho_{\theta}$ , $\displaystyle t_b$ in this case?]

Can anyone please help?

Peter