Up With People, Determinants
June 26th, 2007 by WaltGuest posting at the n-Category Café, Tom Leinster drew everyone’s attention to Sheldon Axler’s manifesto, Down With Determinants! Axler argues when teaching linear algebra, determinants should be avoided almost completely.
This has led to a surprisingly wide-ranging discussion. Lieven Le Bruyn and more recently Masoud Khalkhali have weighed in with defences of the determinant.
June 27th, 2007 at 9:19 am
Determinants are definitely a messy business, especially when you want to work out the eigenvalues of a 4×4 matrix! Steve mentioned a nice way of finding eigenvalues without determinants here: http://www.sixthform.info/maths/?p=160
June 27th, 2007 at 3:02 pm
I am torn here…
On the one hand, in general under most hypotheses, I prefer a coordinate-free approach. On the other, I don’t remember learning the Fundamental Theorem of Algebra until I took Abstract Algebra, which was WAY after I took Linear Algebra.
In the UW numbering system, maybe the standard way speaks to the 307 me, and the other speaks to the post 404 me?
maybe the real problem is that there doesn’t appear to be a supp on the set of instruction orderings?
June 28th, 2007 at 3:44 pm
Determinants that are these nice (IMHO) things that exist for general reasons and have nice properties. I don’t see what’s wrong with them. The problem is when you find yourself computing determinants. Most times when people I know have resorted to computing determinants there has been a much easier way. But don’t be down on determinants just because they’re messy to compute, and you probably don’t need to compute them anyway. That’s like being down on the real numbers just because long division is messy. The problem is long division, not the reals.
June 29th, 2007 at 1:49 am
I think I was first instroduced to determinants and eigenvalues in high school. Sure, they could be a pain to calculate but I always liked them because they had cool properties.
And slightly off topic: I also remember, in my first undergrad linear algebra class, being introduced to norms. After having just done inner products, my reaction to norms was: YUK! This was before I’d heard about metric spaces and I think it wasn’t until grad school before I got over my first impression and learned to appreciate them.
July 5th, 2007 at 1:11 pm
I’m with Axler, 100%. I remember the first time I saw the determinant (of a 3×3 matrix) in my Algebra II course in high school, asking the professor where it came from; I still haven’t gotten a straight answer! Teaching the concepts of linear algebra that don’t require determinants, using determinants, is cheating the students of the opportunity to gain experience using the conventional, less quick, but more transparent and important to master linear algebraic methods. For that reason, Axler’s book is one of my favorite linear algebra texts.
Back to the question of the motivation for the determinant. Historically, where did it come from? I’ve heard the 2×2 came from investigating the solvability of a 2×2 matrix, but that’s far from the whole ‘alternating multilinear form’ definition. Is there a ‘natural’ origin?
July 5th, 2007 at 3:42 pm
Alex, to a large extent you really have to remember that the idea of doing things in a coordinate-independent fashion is relatively recent. Just look at those benighted physicists still juggling tensor indices after all these years.
Back when every vector space had a given basis as a matter of course, determinants were the way to determine unique solvability, and they mirror the use of Gauss-Jordan elimination to solve systems, which encodes what we’d now call a change of basis in the target space.
July 12th, 2007 at 6:57 am
Alex, to answer your question (and expand on what John said):
One way to think of where the determinant comes from is Gaussian reduction. Take a 3×3 matrix you are using to solve for three simultaneous equations (x, y, and z). If you write explicitly the equations (a, b, c, etc for the entries of the matrix), the denominator of each will be the determinant.
Because one can’t divide by zero, when the determinant is zero, a unique solution doesn’t exist.
If you want try the algebra on your own, you’ll find that you don’t have to work all the way through the reduction to get the determinant — it comes out naturally of the steps leading to the last row. From there you can backtrack and get an idea of where the composition of the determinant comes from.