...Prove Their Worth...

"Problems worthy of attack
prove their worth
by hitting back." - Piet Hein

A kind of running diary and rambling pieces on my struggles with assorted books, classes, and other things, as they happen. You must be pretty bored to be reading this...

Monday, October 20, 2003

Ooh. Blogger's updated their interface. Shiny. And sorta blue. Hmm.



I pulled a stupid in my math homework this week, but it's already turned in. The assignment was to show that the group of non-zero complex numbers under multiplication is isomorphic to the group of two-by-two real matrices of the form ((a,b),(-b,a)), or something like that. In any case, the 'obvious' approach is to write an arbitrary complex number z as z=a+i*b, and then just stick the a and b into the matrix, and then show that this gives a one-to-one and onto homomorphism, and we're done.



I was too stupid to do it that way. The trouble is, when someone tells me to think of a complex number, I don't think a+i*b. Instead, I think r*exp(i*t), mainly because this seems to be more useful a lot of the time, and it's the form I've use all the time in physics and in my other math pursuits. I also happen to know from previous reading that the group of unit complex numbers, i.e., things of the form z=exp(i*t), are isomorphic to the rotation group in two dimensions. (U(1) is isomorphic to SO(2,R)) Visually, multiplication by a unit complex number is just a rotation of the plane (thank you, Needham's _Visual Complex Analysis_). So this makes sense. Anyway, I set out to first prove that U(1) is isomorphic to SO(2,R), and then applied that to the actual problem, which can be done by factoring out the square root of the matrix determinant to get an orthonormal matrix times some scaling factor, which is what's equivalent to a general complex number z=r*exp(i*t). This was actually kind of fun to do, and it's a generally cool proof, but it was way, way more work than was actually called for, and the grader may not appreciate having to read through a page-long argument where a few lines would have sufficed. Ah well - I'm pretty sure my proof is right. It's just stupid overkill.



I wonder if I can ever get back intro the habit of updating this thing on a regular basis. Heh. And this page needs to be redesigned, bad. The CSS sucks. Maybe later.

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