High-level linear algebra book

What follows is a substantially edited version of a 25 August 2001 k12.ed.math post of mine.

There are typically 3 different levels of linear algebra that can be found at American colleges and universities. [I'm restricting myself to America because I don't know much about the situation in other countries.]

1. The first level is what is often called elementary linear algebra. This is usually taken by 2nd year undergraduates after they have completed the second or third semester of the standard elementary calculus sequence. However, depending on the college, quite a few 1st year and/or 3rd-4th year students might also be in the class. [In each of the two linear algebra classes I taught during the Spring 2000 semester, over 50% of the students were 1st year students.] I assume this is not the level you're interested in and I'm only including it for completeness.

2. The second level is a course typically taken by upper level math, physics, and (sometimes) engineering students. At some colleges and universities, students may elect to skip the first level linear algebra course and begin with this level. [This was the case where I did most of my undergraduate work. We used Hoffman/Kunze and, when I took the course, there were 5 2nd year undergraduate students (including me) in the course and none of us had taken the lower level linear algebra class.] Texts that would be appropriate for this level are:

Paul R. Halmos, Finite-dimensional vector spaces

Kenneth Hoffman and Ray Kunze, Linear Algebra

Gilbert Strang, Linear Algebra and its Applications

Sheldon Axler, Linear Algebra Done Right

3. The third level is graduate level linear algebra. In many universities the Hoffman/Kunze text above is used (or at least it used to be used), but in these cases the first three chapters are usually covered very quickly (if at all) in order to devote more time to the 2nd half of the text. It is also common for graduate level linear algebra to be incorporated into the 2-3 semester graduate algebra sequence. For example, when I was a student two of the more widely used algebra texts were Lang's Algebra and Hungerford's Algebra, and each contains a substantial amount of linear algebra. Listed below are a couple of "stand-alone" texts for this level. I've had Jacobson since the early to mid 1980s and Brown's book since 1989 or 1990. Brown's book is definitely more modern, but if you're serious about the material, you should at least look at a copy of Jacobson's book (in most U.S. college and university libraries) from time to time. Without knowing anything more about you than what you wrote in your question, I would guess that Brown's book is the best for what you're looking for.

William C. Brown, A Second Course in Linear Algebra

Nathan Jacobson, Lectures in Abstract Algebra. Volume 2. Linear Algebra [See also Dieudonne's Bulletin of the AMS review of Jacobson's book.]

(TWO MORE "THIRD LEVEL" TEXTS, ADDED A YEAR LATER)

Werner Hildbert Greub, Linear Algebra

Steven Roman, Advanced Linear Algebra

Having read almost every book mentioned here, I can tell you that "linear algebra done right" by Axler is hit or miss. Over half the problems in ch.3,6,7,8 are impossible to answer the others are quite simple. The material covered is ideal, but has no worked examples and offers no computational method. Finite dimensional vector spaces by Halmos is a short read with mediocre problems and the book is from the 40's and mostly outdated.

S. Winitzki, Linear Algebra via Exterior Products (free book, coordinate-free approach).