6.4 Inverse of Isometries (Basic Mathematics)

Can we compose any isometry to obtain the identity? Much like in addition and multiplication, you can. We won't prove that all isometries are invertible (see section 6.5 to do that for yourself), but we will expose how finding the inverse allows us to uniquely identify isometries, as well as exhaustively examine them.

Thank you!

[PythonDad]

I started working through this book myself after seeing a lot of discussion suggesting it's a great way to learn to write proofs, and I felt like I was getting somewhere until this very section. I feel as if I've hit a wall at this point. For the exercises that are making proofs for the last three sections of chapter six I wrote something for them which doesn't sound like complete nonsense, but I don't really have the same kind of self-assurance I felt for proofs from earlier. Should I simply go back to the theorems which are proven in the text and try to glean what I can in both reasonable assumptions and techniques for proving and then do those on the things not proven in the text? I specifically finally used this book because all the "proof" books seem to be "bags of tricks" and my intuition on an approach to take, in general, is very bad. It's been rather disheartening, but I feel that if I can master this, my confidence going forward in study will probably improve.