Linear algebra, and more than one such book.

IMHO long and still the best linear algebra book is

Halmos, Finite Dimensional Vector Spaces (FDVS).

It was written in 1942 when Halmos was an "assistant" to John von Neumann at the Institute for Advanced Study. It is intended to be finite dimensional vector spaces but done with the techniques of Hilbert space. The central result in the book, according to Halmos, is the spectral decomposition. One result at a time, the quality of von Neumann comes through. Commonly physicists have been given that book as their introduction to Hilbert space for quantum mechanics.

But FDVS is a little too much for a first book on linear algebra, or maybe even a second book, should be maybe a third one.

Also high quality is Nering, Linear Algebra and Matrix Theory. Again, the quality comes through: Nering was a student of Artin at Princeton. There Nering does most of linear algebra on just finite fields, not just the real and complex fields; finite fields in linear algebra are important in error correcting codes. So, that finite field work is a good introduction to abstract algebra.

For a first book on linear algebra, I'd recommend something easy. The one I used was

Murdoch,
Linear Algebra for Undergraduates.

It's still okay if can find it.

For a first book, likely the one by Strang at MIT is good. Just use it as a first book and don't take it too seriously since are going to cover all of it and more again later.

I can recommend the beginning sections on vector spaces, convexity, and the inverse and implicit function theorems in

Fleming, Functions of Several Variables

Fleming was long at the Brown University Division of Applied Math. The later chapters are on measure theory, the Lebesgue integral, and the exterior algebra of differential forms, and there are better treatments.

Also there is now

Stephen Boyd and Lieven Vandenberghe, Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares

at

http://vmls-book.stanford.edu/vmls.pdf

Since the book is new, I've only looked through it -- it looks like a good selection and arrangement of topics. And Boyd is good, wrote a terrific book, maybe, IMHO likely, the best in the world, on convexity, which is in a sense is half of linearity.

Some course slides are available at

http://vmls-book.stanford.edu/

For reference for more, have a copy of

Richard Bellman,
Introduction to Matrix Analysis:
Second Edition.

Bellman was famous for dynamic programming.

For computations in linear algebra, consider

George E. Forsythe and
Cleve B. Moler,
Computer Solution of Linear Algebraic Systems

although now the Linpack materials might be a better starting point for numerical linear algebra. Numerical linear algebra is now a well developed specialized field, and the Linpack materials might be a good start on the best of the field. Such linear algebra is apparently the main yardstick in evaluating the highly parallel supercomputers.

After linear algebra go through

Rudin, Principles of Mathematical Analysis, Third Edition.

He does the Riemann integral very carefully, Fourier series, vector analysis via exterior algebra, and has the inverse and implicit function theorems (key to differential geometry, e.g., for relativity theory) as exercises.

All of this material is to get to the main goals of measure theory, the Lebesgue integral, Fourier theory, Hilbert space and Banach space as in, say, the first, real (not complex) half of

Rudin, Real and Complex Analysis

But for that I would start with

Royden, Real Analysis

sweetheart writing on that math.

Depending on the math department, those books might be enough to pass the Ph.D. qualifying exam in Analysis. It was for me: From those books I did the best in the class on that exam.

Moreover, from independent study of Halmos, Nering, Fleming, Forsythe, linearity in statistics, and some more, I totally blew away all the students in a challenging second (maybe intentionally flunk out), advanced course in linear algebra and, then, did the best in the class on the corresponding qualifying exam, that is, where that second course was my first formal course in linear algebra.

Lesson: Just self study of those books can give a really good background in linear algebra and its role in the rest of pure and applied math.

No joke, linear algebra, and the associated vector spaces, is one of the most important courses for more work in pure and applied math, engineering, and likely the future of computing.