Back in the day, I didn’t carefully read the proofs and statements about change of basis of a vector space. New basis is P times the old one for some invertible P? Fine. I’d skim through the proof with the usual tedious subscripts, tiny little i’s and j’s all over the place, trust the author knew what he was doing, and leave it at that.
And I got into a hopeless muddle. As the book went on, sometimes the author would toss around a P, sometimes a P inverse. What’s that about? P is invertible, right? So why throw around the P inverse, when you could have renamed the inverse Q, say, and left it at that? To make things worse, the authors seemed to contradict themselves, too. Sometimes they’d say it was P that counts, sometimes P inverse. I mean, what gives?
Finally I got it. A change of basis is talking about two new things. There’s the new basis vectors, and on top of that there’s the new coordinates of other vectors in terms of the new basis vectors. The Aha! moment was when I realized that if the new basis vectors are related to the old ones by P, then the new coordinates are related to the old ones by P inverse.
And it makes sense, in a way. Since the new basis vectors complicate things [by using P], then the new coordinates better un-complicate them [using P inverse] if we want to get the same vector.