Wow!

So just guess which branch of mathematics played a role in the pure mathematical research I was working on this afternoon?

Hmm? Any guesses? I'll give you three, and the first two don't count.

Here's the problem. I've got a set of distinct integers a_i, 2 or larger, arbitrarily many of 'em, maybe something like: {2,4,5,8,13}. Now you get to pick any positive number a (doesn't have to be an integer) lying strictly between the smallest element of the set and the biggest. In our example above, I could take 5.4, maybe.

The question is: how many different ways can I pick "weights" x₁, x₂, x₃, and so forth, one for each element of my original set, so that each weight lies on [0,1], the sum of the weights is 1, and the sum a₁x₁ + a₂x₂ + ... +a_nx_n is the number a? Above, for instance, we'd get to pick 5 weights, and we'd have to get the sum from the previous line to add up to our chosen value of 5.4.

I'll leave it as an exercise for the reader (a common math ploy!) to provide the details, but I'll cut to the chase: you can turn this problem into a linear system, and from there into a matrix, with 2 rows and as many columns as you have numbers a_i in your set. Since you can have arbitrarily many elements in that set, you can end up with boatloads of columns (yes, "boatloads" is a technical term). And as soon as you've got more than 2 columns...what happens? I'll let you work it out.

Cool. This has some pretty heavy-duty implications in the land of graph theory.

I was so excited about figuring this out this afternoon that I went across the hall to the Math Lab, found Kaytlynne and Theophila working hard on homework from their other courses, and bugged them about the cool way linear algebra had worked itself into my work.

Neat, huh?

Tomorrow we'll get a few more folks from the class up at the board. Nadia and Imogene did a great job in getting us started on mapping out crystal structures. We'll pick it up from there.