I'm trying to understand this proof that every origin-preserving rigid motion can be represented as left-multiplication by a matrix. The proof is given in Artin's "Algebra", pp 127-128. I've posted an image of the relevant text here:
The paragraph with the blue line proves that "if m is a map that (1) preserves dot product and (2) fixes the basis vectors, then m is the identity map". That's fine.
In the next paragraph, he assumes that m preserves dot product, builds the matrix A, and asks us to consider the composite map A-1m. It's clear to me that this composite map preserves dot products. But then he declares without justification (underlined in red) that it also fixes the basis vectors. I can't see why.
Since A is orthogonal, A-1 is just AT, right? But even using that, I can't seem to prove that A-1m applied to ei is ei.
What am I missing? Thanks.