Advanced Linear Algebra

This is the main page for Advanced Linear Algebra, Spring 2021. You can find links to all the materials here.

There are three main channels of communication for this class: the course channel on the Bennington Math Slack, the synchronous Zoom classes, and this course webpage.

  • All announcements (including posting of problem sets, scheduling of progress meetings, due dates, corrections, etc.) will be posted on the Slack. I also encourage you to ask questions on the Slack (both of me and of other students), and to have class discussions there.
  • The class time is the main place where the content of the class happens, as well as discussions and collaborative work; please try to attend all classes. Recordings of the Zoom classes will be posted on an unlisted YouTube channel (link in the Slack) in case you miss any.
  • Finally, all problem sets, readings, and other course materials will be linked on this course webpage. Additionally, time permitting, I may write up some lectures or supplementary materials, which I will also link to from here.

Lectures

Problem Sets

  • Problem Set 1 (introducing the Projection Problem and least squares)
  • Problem Set 2 (on least squares)
  • Problem Set 3 (giving a solution to the projection problem, and developing material on vector spaces and bases)
  • Problem Set 4 (on linear transformations, dual vectors, and bases)
  • Some Applications of Eigenvalues (Problem Set 5) (as the title says. This problem set is actually more longer-form problems and projects, so you can just pick one or maybe two. It is both lecture and problem set; although I don’t expect you to do all the problems, I recommend at least reading through all of them.)
  • Problem Set 6 (not so much problems, as a list of suggested theory topics for further reading, which you can work through; see the problem set for an explanation)

For those who use LaTeX: a zip file with all the LaTeX source for problem sets so far.

Resources

The best free textbook that I’ve found for the subject at this level is:

Linear Algebra, by Jim Hefferon.

It may be helpful to search the table of contents for a topic we have been doing. He has pretty good coverage of much of the theory (though he doesn’t seem to get to fancier things like spectral decomposition and dual spaces), and he covers several of the applications we have been talking about.

Another free book that looks like it might have nice features is

A First Course in Linear Algebra, by Robert Beezer.

One nice feature is a chapter on doing computations in Sage/Python. (Many other books have computational chapters in MATLAB, which is a bit old-fashioned at this point.) It also has a chapter on representations, which might be of interest to the Abstract Algebra folks.

Many linear algebra books (the above included) are kind of long-winded. A nice introduction that looks brief and to the point is

Linear Algebra, by Carter, Tapia, and Papakonstantinou

That book is nice and straightforward on the basics, but it doesn’t get to anything fancy.

For some of the fancy topics like dual spaces, this free book looks very cool:

A Course of Linear Algebra and Multidimensional Geometry, by R. A. Sharipov (pdf, arxiv link)

I admit that I haven’t had time to read it yet, but it seems pretty close to my own approach.

I have so far listed books that are freely available. Here are some which might be harder to get a hold of, but which may be of interest for further reading. I will list them in increasing order of sophistication.

  • Elementary Linear Algebra with Applications, by Kolman and Hill. I used this book for planning the basic structure of the class, and for its applications. I also used it when I taught the intro. It doesn’t go as far as I would like, but it’s my favorite intro book. There should be copies in the reading room and at the library. The current edition is stupidly expensive, but you often can find older editions (which are mostly the same) for much cheaper.
  • Linear Algebra by Serge Lang. Clear and to the point, with a clean modern looking presentation.
  • Lectures on Linear Algebra, by I.M. Gelfand. This book is more modern and sophisticated. It is very close to the approach I am taking, and I think when I teach this class again I will use it as a textbook. It is quite terse, which can be good and bad. It has a slightly old-fashioned language, but the approach is actually much more modern than newer books. It is a Dover, so it is quite inexpensive.
  • Linear Algebra Done Right, by Sheldon Axler. This book has a better approach to eigenvalues than the classic one with determinants, and some other interesting novelties. A great book if you like the style.
  • Finite-Dimensional Vector Spaces, by Paul Halmos. This is THE classic book on vector spaces, which every math major should read at some point. It is the classic development of all the core theory. It is now available in an inexpensive Dover reprint.
  • Linear Algebra and Multilinear Algebra, by Werner Greub. Unfortunately quite expensive, but the best books for being clear on modern theory, particularly for quotient spaces and for tensors.
  • Geometric Algebra by Emil Artin. A beautiful, beautiful book, by a gifted expositor and student of Emmy Noether. Despite being written in 1957, it’s a more modern and insightful approach than any of the texts above. Covers a lot more than linear algebra, so the linear algebra is brief. Now available in a Dover reprint.
  • Galois Theory by Emil Artin (pdf). Again, a beautiful book. Condenses all of linear algebra into 20 pages, then applies linear algebra to the theory of fields (Galois Theory). Whole thing is 82 pages, in Dover.
  • Applications of Tensor Analysis, by A. J. McConnell. A great introduction to tensor analysis, in the old style. Parts I and II cover linear algebra and geometry; Parts III and IV go further than we do, applying this to differential geometry and physics.
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