The first three days of the quarter have been interesting. The most surprising thing I have found is how much free time I have. All morning yesterday, I had to convince myself that I didn’t actually have a class to go to. I’ll get used to it soon enough.

Algebra might be pretty tricky this quarter. The professor said we have to know all the proofs well enough so that we can recreate them on the board if she were to ask one of us to do so. That’s pretty scary for me.

Complex analysis should be easy this quarter. We have to write a paper, but that’s it. We had that last year one quarter in real analysis (more or less), and that was good because I learned stuff outside of the normal course material.

Since I have so much free time, I have been sitting in on a few classes. I went to combinatorics on Tuesday even though I read most of the textbook (Concrete Mathematics by Graham, Knuth, and Patashnik) in high school. As a result, I knew all the things he told us. I think I scared some people by knowing things like the derangement of 4. But that was fun, and I’ll continue going.

I also went to the advanced linear algebra seminar, which replaces the functional analysis seminar because Putinar is away this quarter and so there is no funding for it. Here’s a neat proof that we did today:

Theorem: F_(n+1)F_(n-1)-F_n^2=(-1)^n.

Proof: Consider the matrix A=(1 & 1 \\ 1 & 0). In general we have A^n=(F_(n+1) & F_n \\ F_n & F_(n-1)). (This is easy to show by induction.) Since det(A)=-1, we have det(A^n)=(-1)^n=F_(n+1)F_(n-1)-F_n^2, as desired.