I am teaching an undergraduate course in linear algebra this fall. I am dissatisfied with most existing textbooks, and indeed with the way in which this subject is usually taught. I hope to find a textbook that has many or all of the following characteristics:
It emphasizes linear transformations as the central object of study. This, in my view, is what the subject is about, and I think a linear algebra course should start with them (perhaps only in R^2 or R^3) from the very beginning.
Instead, most books start with solutions to systems of linear equations, which I personally believe is of secondary importance. Moreover they use "augmented matrices" which (in my mind) obscure the relationship between matrices and linear transformations. I would prefer to never write down a matrix which does not represent a linear transformation in the usual way.
It takes a heavily geometric approach. Linear transformations can be visualized and drawn, and I think a linear algebra course which doesn't emphasize this is selling itself short.
As an example, I think the following would make a great homework and exam question: Shown are a pair of coordinate axes and a cartoon drawn on the plane. A 2x2 matrix is given. Sketch the image of the cartoon under this linear transformation.
It de-emphasizes the carrying out of algorithms by pencil and paper. I have memories of row reduced echelon forms, pivots, etc., etc. but as a working mathematician, none of this really stuck. I memorized the techniques, regurgitated them for the exam, and although I could reconstruct them if needed I don't really remember them.
It de-emphasizes axiomatics. In my opinion it is uninspiring to see a list of axioms for a vector space and to prove statements like "If v + 0 = 0, then v = 0". This goes especially since my students won't have the mathematical background or maturity to appreciate some of the more eclectic examples of vector spaces, or why you would be interested in working over any field other than R or C. They are also somewhat unlikely to have seen examples from any area of math where intuition leads you astray.
It really tries to teach the "meaning" of a matrix. To my embarrassment, it was after I finished my Ph.D. that I realized the columns of a matrix are just the images of the basis vectors. This was probably said in the book I used as an undergrad, but not really emphasized. Similarly, when we solve linear equations we write each equation as a row vector rather than a column vector because row vectors in fact represent "covectors", i.e. elements of the dual space.
I don't want to go overboard with this (i.e., to the extent of introducing category theory), but I do want to present some of the notions of duality that are at the heart of linear algebra and more advanced mathematics.
It is a good book overall.
It doesn't cost $300 or whatever exorbitant price publishers are charging these days.
I should mention that my university offers "applied" (using Strang) and "honors" versions of linear algebra, and that the course I'm teaching is neither.
Based on the preferences I've described, does anyone have books to recommend? Thank you very much.