I noticed some egregious typos in my notes, so I’ve updated the file.

This semester, the first homework assignment is going to be based on our discussion of rotations in 2D. There were a few claims I made without proof (exercise numbers are from my notes.):

1) Prove that the rotated vector components really are supposed to be what I claimed. (Ex. 2.1.)

2) Prove that the length of the vector is unchanged by rotations. (Ex. 2.2.)

3) 2D rotations commute. Note that this is not true for higher dimensions. (Ex. 2.3.)

We didn’t discuss this in class, but there are an infinite number of complex representations of the 2D rotation group. You can work these out by mapping the 2D vectors to complex numbers. For example, you can let the x-basis vector map to 1 and the y-basis vector map to i. After doing that, any other 2D vector maps to a complex number. A fun exercise is to see that our vector rotations correspond to multiplication by a phase exp(i theta). (Ex. 2.4.)