So you want to be a non-commutative geometer, Part I: Preliminaries

OK, these are just my random notes on my studies of non-commutative geometry. I hope that someone who understands only vector calculus (and perhaps linear algebra and most certainly abstract index notation — tensors and all that jazz) alone could look through this series and become a non-commutative geometer.
So, before we begin, we should probably [...]