Posts Tagged ‘linear algebra’

The exchange lemma and Gaussian elimination

October 3, 2007

Thanks to this comment, I have finally decided to try to understand in what sense Gaussian elimination and the Steinitz exchange lemma are “basically the same thing”. It’s not at all hard to spot similarities, but it seems to be a little trickier to come up with a purely mechanical process for translating proofs in one language into proofs in the other.

It might be of some interest to know how I approached this post. Rather than working everything out in advance, I started with an incomplete understanding of the connection. I had thought about it enough to convince myself that I could get to the end, but found that as I proceeded there were a few surprises, and the eventual connection was not quite as close as I was expecting. (Actually, this paragraph is slightly misleading. I am writing it while in the middle of writing the rest of the post. I’ve had a few surprises, and though I am fairly sure I’ll get to the end I am not quite sure what the end will look like. [Note added after I’d got to the end: it was nothing like what I expected.]) (more…)