A friend of mine in graduate school used to joke about the existence of a “secret linear algebra curriculum,” i.e. a body of knowledge from linear algebra that all graduate students are supposed to know but no one will ever teach them (in fact, no one will even tell them what they should know, hence the secret part of it!). This resonates with me very strongly; in particular, I had a standard mathematics curriculum for an undergraduate in the US, which means I had exactly one semester of linear algebra which really meant a semester of matrix algebra that almost sorta kinda got to eigenvalues and eigenvectors and certainly didn’t cover what math students should really know.
In subsequent classes, my professors assumed I and my peers knew all about linear algebra, when in fact we didn’t. This continued all the way until graduate school, at which point this became a serious problem. I have often told students that it wasn’t until I learned linear algebra that I was able to prove anything worthwhile.
While many people, including myself, will say that ultimately it falls upon the student to make sure they learn what they need to know (no professor can force a student to learn if they don’t want to), the big problem with linear algebra is that many times it is hard to figure out where to look to find out what you should know! Hence, in this post I want to mention the two references that contain what I consider to be the core of the secret linear algebra curriculum. While these articles alone don’t contain all the information that students (grad and undergrad) need to know, they summarize the subject well and put linear algebra in the correct perspective.
The first article is
The Fundamental Theorem of Linear Algebra
Gilbert Strang
The American Mathematical Monthly, Vol. 100, No. 9 (Nov., 1993), pp. 848-855
(this is available on www.jstor.org if your institution has a subscription).
The second article is
Down With Determinants!
Sheldon Axler
American Mathematical Monthly, Vol. 102 (1995), 139-154.
These articles approach linear algebra from totally different perspectives: Strang is matrix based and stays in real space, while Axler is coordinate-free and works over the complex field. However, by reading through these two overviews, students can see how all the pieces fit together, and then they can get on with the business of learning and proving.
