Fun little rebuses, which came to me while listening to my brother speak in math.

This is wonderful. I’ve seen it time and time again, and it never ceases to bedazzle me. Say we have a transform between two coordinate frames. In other words, we have the transform that takes frame A and turns it into frame B (through rotation, scaling, translation… whatever. Anything affine).

That, then, is the transform that takes points from frame B and puts them into frame A! To recap: it transforms frame A into frame B, but it transforms points in frame B to points in frame A! Isn’t that cool? Wouldn’t you expect it to be the same direction in both cases?

If your name isn’t Zoe, you might not believe me, so I’ve sketched out a couple of 2D examples just to prove that this is really a thing. (For the sticklers out there, I use the word “prove” in a very informal sense.)