After six weeks of teaching, I’m still having a great time. In fact, it is only getting better! My students are great. The material is getting more interesting. Everything is beautiful.
My linear algebra class is studying determinants. The first section on determinants in the book that I use is strictly on the method of cofactor expansion, and gives no properties of the determinant outside of how to calculate it. I assigned the True-False problems from book without giving careful thought to the tools available in each section. So of course I’m leading up to how I assigned a problem that I couldn’t solve with those tools. Here it is. True or False: If A is a square matrix in which all minors have the same value, then det(A)=0.
Okay, so the answer is true, which you can see in several ways. The easiest way is to consider the adjoint formula for the matrix inverse. The adjoint is the transpose of the matrix whose entries are the cofactors of the original matrix. A inverse is 1/det(A) times the adjoint matrix. If all the minors have the same value, then this supposed A inverse is not invertible, so A is not invertible, so A has determinant 0. But wait! The book has not made any connection between invertibility and the value of the determinant. And the adjoint formula doesn’t come until 2 sections later. So how could you solve this problem without either of these pieces?
The excellent Billy Chan, my next-door office neighbor here, found a very nice proof that doesn’t require the adjoint formula, but does require invertibility… and oh, hooray! I just figured out how to skip the invertibility step. Here’s the idea–create B by replacing the second row of A by another copy of the first row. The determinant of B will equal the determinant of A because if we do cofactor expansion along the first row of B, the minors we encounter are just the second row minors from A, and all these have the same value, so they are in fact the same as the first row minors from A. In any case, B has two identical rows, so is not invertible and has determinant 0. But we don’t need to know that connection if we just write the cofactor expansion of B along the first two rows. The two expressions for the determinant are identical but have opposite signs, so the determinant must be 0.
Whew! So that took two PhDs. Still, I bet that at least one of my students figured it out.
In other good news:
I finished revisions on a paper about the global zeta function of Gauss’ curve (will post soon)!
I get to talk about my maximal curve research at 3 different colleges next month.
Fall in Connecticut is looking like it might be as beautiful as everyone claims.
One of my students spent an hour in my office with me this morning teaching me how to knit socks.
In the making good news from bad news section:
After six weeks with a cone on his head, my dog Arlo has forgotten what it was like before he wore one. Arlo was hit by a car during our first week in Middletown. He got away with what seemed a miraculously small injury–basically a big laceration on his leg. He needed a small surgery to stitch it up, but no bones were broken and there were no internal injuries. However, he has to wear the aforementioned Elizabethan collar to keep him from licking his leg off, which I think that he would do if given half an hour or so. This collar was, for about 5 weeks, the most annoying thing ever for both of us. Arlo kept getting stuck in doorways because he couldn’t comprehend that he needed 3 times the usual clearance to get by something. He would drag it along the wall when he walked, which is VERY LOUD. Yes, all caps loud. Someone should look into recycling e-collars in the form of really annoying instruments! The semi-rigid plastic makes a great resonator and creates almost a roar as it is dragged along any surface. Especially in the middle of the night when my poor dog is restless and gets up and walks around the room once an hour. One day, in the course of dragging the thing around, he got the collar stuck under a chair and managed to get his head out of it. Of course he pulled all his stitches out and reopened the wound. Disaster. The new plan has been to let the wound heal without stitches, with bandage changes at the vet’s office every two days. I have been pretty frustrated with the whole thing, and getting very impatient for the bandage changes to be done. At various times, the end has seemed tantalizingly close. Maybe only another week? Two weeks? It seemed possible that it could be almost over! When?!! Every tentative deadline has passed, though, and the end has continued to slip back out of sight.
But, remember, this was good news. The good news is that I am just not that worried about it anymore. As I said earlier, I think that in the last week Arlo has forgotten what life was like before the cone and now seems perfectly content. And, I mean, really, at least *I* don’t have to wear the cone. And I get to read in the waiting room. Things could be worse.