Tripping up on vector spaces

So it's been a month and this is only my second post. I had originally intended to give a brief description of the field of linear algebra, but I got horribly tripped up in two places. This last month I've been either struggling with these two things or else sticking my head into the sand and ignoring them.

1) Linear algebra is described everywhere as the study of "vector spaces." Well vector spaces strike fear into my heart the way "mathematics" seems to strike fear into the heart of half the country. When I went to college the first math course I took was multivariable calculus, which is filled with vectors, and, presumably, vector spaces. Princeton had three levels of multivariable: regular, advanced, and some kind of theoretical honors thing. I started out in the mid level course, and I had one of the few female professors in the department and one of the few who was a native speaker of English, which should have given me a big advantage.

Well I didn't do very well. I just couldn't process the images of planes and arrows and axes. I got panicked. The professor was so kind and spent a lot of time trying to help and encourage me, but in the end I dropped down a level into the regular course, which was still strenuous but was more routine. (This, for any of my old college friends, is where I had the professor who seemed to own a single pair of pants.)

Anyway, I think this story illustrates two things: that math panic happens to everyone, even people who were raised around math, and that different mathematical fields attract different people. I felt far more comfortable in other courses.

You may wonder here, though, why vector spaces show up in the definition of linear algebra, when I am talking about them in multivariable calculus. Well, this is where my jig is up. Heck if I understand it all yet. I'll get back to you on that.

2. The best way to explain anything is through an example, but to discuss a matrix-related example, I have to draw some matrices. I couldn't for the life of me figure out how to do it in Word. I might just scan some hand drawn ones and upload them.

So now you might never want to read any more posts here. You have seen how unqualified I am to lead you along the linear algebra path. Or, you may say, Hey! She doesn't know anything about this stuff either! And she, like me, is baffled by Word 2007! So maybe we are kindred spirits and I'll be able to understand what she is talking about!

Let's hope for option number two.

Statement of the Problem

My father is a mathematician pretty well known in his field of linear algebra. (I'll tell you what linear algebra is in the next blog post.) He is a professor at the College of William and Mary in Williamsburg, Virginia. He spends most of his time doing research (another blog topic) but he also really likes teaching, as long as it is either at a fairly high level or his students are pretty sharp.

He especially enjoys running and teaching at an Research Experience for Undergraduates (REU) program in the summer. The program is made up of college math students from around the country, and the professors present a bunch of small problems that they find interesting but don't know the answer to. Each student picks a problem and a professor and tries to get results by the end of the summer.

I asked my dad if he could give me a math problem to work on, since he is very good at collecting problems that are narrow in scope and good for beginners. I am a career writer, currently working as a technical writer for an online advertising startup, where I write about our products and software. My first love is creative nonfiction, especially essays. I took linear algebra in college in 1998, and did perfectly well but not brilliantly. I am better and more comfortable at general math than the average American, but have never showed any particular talent at the college level. I am interested in math as a concept, but have never been drawn to problems the way my dad is.

I am pretty terrified of this problem, because I am insecure about my high-level mathematics ability, and truthfully, I would really like to impress my dad. So I have been shying away from it. I thought maybe chronicling my progress on it through a blog, where I will then be responsible to my millions of clamoring fans, might help.

So. Here is the problem. Feel free to work on it yourself, but if you solve it, don't tell me the answer. In the next post I will define all the terms and add a little more information about it from my dad, which he imparted while perched precariously on a stool at a paella restaurant near my apartment in New York City.

Dear Em. Let me try to give a rough description of the problem. A matrix is called totally positive TP) if ALL its minors (determinants of square submatrices) are positive. A weaker condition (TP2) is that only the entries and 2-by-2 minors are required to be positive.(However Shaun and I have shown the remarkable result that some entry-wise power of a TP2 matrix is TP.). Now, it is a theorem that a matrix is TP2 if and only if it is entry-wise positive and all its contiguous 2-by-2 minors (just the ones with consecutive row indices and consecutive column indices) are positive. There are many fewer of these. Now, consider, in place of a conventional (rectangular) matrix an arbitrary array that may take on any rectilinear shape (perhaps a fat "L" or a fat "U", among many). The question is, for which connected shapes does contiguous 2-by-2 minors positive (in addition to positive entries) imply that all 2-by-2 minors are positive? Note that I get a 2-by-2 minor ANY time I have a square 2-by-2 submatrix, even if there is space in between.