Advanced Linear Algebra courses in graduate schools

After studying general a linear algebra course, how would an advanced linear algebra course differ from the general course?

And would an advanced linear algebra course be taught in graduate schools?

Solution 1:

At my [undergraduate] university [which was University of Cincinnati, at the time of this post], the first linear algebra sequence is taught to sophomores. It is mostly computational. Everything takes place in the reals and complex numbers. The class begins with row reducing and culminates with finding determinants and eigenvalues. I don't remember which book we use for this but it's terrible and the class is very easy.

Later, students are encouraged to take "abstract" linear algebra, which focuses on abstract vector spaces (though they are all assumed to be over fields of characteristic $0$), inner product spaces, quadratic forms, proving the spectral theorem, and culminates with Jordan canonical form and the theory of convex sets. For this we use Lang's linear algebra. More emphasis is placed on the spectral theorem than anything, with Jordan form and convex sets only if the class moves fast enough so there's time.

Finally, after a student has taken the senior level abstract algebra sequence (featuring the basics of groups, rings, and fields), he may elect to take the graduate algebraic structures class, in which module theory, more advanced ring theory, and some representation theory are covered. For this class we use Dummit and Foote (and whichever other books we feel like). [At my incumbent university, University of Florida, basic ring and module theory is done in the first year graduate course, also using Dummit and Foote. Multilinear algebra (tensors) and more advanced ring theory are covered during spring semester of second year graduate algebra, which concurrently uses Lang, Hungerford, and Matsumura.]

I have heard that other universities offer graduate courses strictly in advanced linear algebra. An example of a book they may use is Roman, which I have used as a reference many times and I must say I like very much.

Solution 2:

First of all, it's not clear what an advanced course in linear algebra at either the undergraduate or graduate level consists of. It really depends on what the first course consists of and this varies enormously from university to university depending not only on the background and career paths of the students, but the aims of the instructor. It can be a largely applied course where rigorous theorems about linear transformations and abstract vector spaces are either largely avoided or downplayed, such as those based on Gilbert Strang's textbooks. It can also be a highly abstract course where applications are barely mentioned at all and the fine theoretical structure of finite dimensional vector spaces is developed in full detail, such as Axler's, Halmos' or Hoffman/Kunze's textbooks. And there are textbooks which try to steer a middle course between the 2 extremes, developing both theory and application in more or less equal measure. The classic example of this kind of course is Charles Curtis' textbook.

I'm quite sympathetic to the last kind of textbook - finding both the theoretical and applied sides of linear algebra to be of equal importance in developing the subject in it's fullest utility for both mathematicians and scientists.

Then again, it's not that cut and dried often, either - often actual courses in linear algebra resist such simple classification - therefore, advanced courses to follow such classes up will be even more difficult to construct. Gilbert Strang's justly famous course at MIT, for example, is a course built around the applications of linear algebra to real world problems. But it's hardly a plug-and-chug, mindless algorithm course: Strang analyzes each application and algorithm, as well as the theory behind it, thoroughly. But at the same time, it's not really an abstract mathematics course the way we describe it-the deep theorems and proofs of linear algebra, while not ignored, are not really the core concerns of the class. For Strang, the abstract theory of linear algebra is really the domain of an abstract algebra course. (Indeed, Strang's course is partially designed to provide a mastery of the computational aspects of linear algebra needed for MIT students to go on to effectively study modern algebra in Micheal Artin's equally famous course!). However, this is MIT we're talking about-hardly your average program with average mathematics majors.

In my experience, what most people mean when they say "advanced linear algebra", they mean the abstract theory of linear operators in the context of modern algebra. At most universities, this material is covered in serious abstract algebra courses at either honors undergraduate or first-year graduate level. This means the study of R-modules over commutative rings in the special case where R is a commutative division ring i.e. a field. This means modules, algebras over R, submodules, R-module maps, product spaces, the Jordan-Holder theorem, tensor products, dual spaces, free modules and perhaps some elementary homological algebra. As I've said, this is usually covered in the student's first substantial year-long algebra course, at either the undergraduate or graduate levels.

Also in my experience, there usually isn't a separate "advanced linear algebra" class for students at the advanced undergraduate or graduate level. But there are exceptions. For example, Peter Lax's linear algebra book is based on a graduate course on the subject that he's taught there for many years designed to bring incoming graduate students at NYU who are weak in linear algebra up to speed for a second year functional analysis course. Lax became frustrated with the anemic skills in basic linear algebra most graduate students at NYU had and designed this course to rectify this very damaging lacuna in thier training.

If you're looking for a text that focuses purely on the abstract theory of linear operators, the best book is probably Module Theory: An Approach To Linear Algebra by T.S. Blythe. THE book on the subject and sadly, out of print and hard to find. Here's to Dover republishing it.

Hope that answers your question.

Solution 3:

Different universities will teach different things under the heading "advanced linear algebra", and at different levels. I would suggest you go to a few university websites and see what they have on offer and what the contents are.

At my university, we teach a 3rd-year undergarduate course which is half Galois Theory, half Numerical Linear Algebra. Macquarie University, Math 338.

Solution 4:

To answer your question succinctly, a first course on linear algebra should cover the basic computational tools: row reduction, determinants, and eigenvalues. A more advanced course should force the students to come to terms with more abstract language (vector spaces over an arbitrary field), and it should contain a sophisticated treatment of the spectral theorem. In an ideal world it would also introduce multilinear algebra and/or the canonical forms, but these topics are often reserved for graduate courses (often in the context of rings and modules).

In practice there is quite a large gap between advanced undergraduate and graduate algebra. This can lead to strange circumstances wherein students first learn about tensor products in a differential geometry course or the Jordan canonical form in a number theory course. I recommend that undergraduates take as much linear algebra as possible since good graduate programs will often assume that students know more linear algebra than they do in practice.