I am reading Papantonopoulou: Algebra Ch 14 Symmetries. I am seeking to fully understand Theorem 14.21 (see attached Papantonopoulou pp 459 -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 axis.

Then on Page 463 he states Theorem 14.21 as follows:

14.21 TheoremAn isometry S of or can be uniquely expressed as where i = 0 or 1

The proof (see attached) proceeds as follows:

Proof: To prove existence, let b = S(0). Then fixes the origin and so for some A O(n) ... ... etc ... ( see attached for definition of )

I need some help with this first line of the proof!

Restating the proof explicitly in we have S(0,0) = ( )

i.e. = ( )

Thus

But ... why does ???

Presumably this is because Papantonopoulou defines rotations as about the origin (0,0) and reflections are defined to be about the X-axis or axis. (see definitions of rotations and reflections at bottom of page 462 - attachment)

A further question is this ... what is Papantonopoulou actually trying to do when he starts his proof with "To prove existence, let b = S(0)"? What is his strategy here? What is he tryinh to do - I am somewhat lost regarding the overall point of this!

I would be really appreciative of some help here - or at least confirmation of my reasoning.

Peter