Wherefor and Why

My student's assignment for today involved some rather tedious calculations. She handed me the write-up, and I scanned it - one of those situations where it's easier to verify that an answer is correct than to actually get the answer. No problems.

Then we went on to the topic of the day, The Care and Feeding of Involutions. I showed her a bunch of tricks for dealing with involutions (transformations whose square is the identity transformation): projecting to eigenspaces, computing the dimensions of eigenspaces without computing the spaces themselves, tricks with commuting involutions, stuff like that. In passing, I pointed out how they could be used to simplify those tedious calculations. Her response was pleased - "Yeah, that's much better" - but there was a hint of asperity in it. (Unspoken: "Why didn't you show me this before assigning me that problem?" "So you'd appreciate it!")

There is a psychological element to the business....