I've used Axler's book in the past as textbook in linear algebra courses I've taught, and I'm familiar with its content. The book is essentially self-contained, so yes, your background should be enough in terms of prerequisites and pace.
More important is certain "mathematical maturity", as the book is fairly theoretical rather than computationally oriented, and you need to be comfortable with proofs. Whether the book will prepare you for later courses is hard to say exactly. You probably want to complement it with another text where there is more emphasis on computational aspects of matrix theory.
In particular, topics that numerical analysts consider fundamental, such as the singular value decomposition of a matrix, or the $QR$ (Francis) algorithm for computing eigenvalues are not really treated by Axler. The key contribution of the book is the development of the basic theory without needing to appeal to the determinant. I believe this is important to actually understand the content of some of the key results. That said, the determinant is an important tool in computations, and this is something Axler does not treat in appropriate depth.
If you are interested in applications of linear algebra beyond the real or complex settings, applications where now the underlying field may be finite, you will need a different book, since Axler does not mention these topics. A sugestion here is Linear Algebra Methods in Combinatorics. With Applications to Geometry and Computer Science, by László Babai, and Péter Frankl. The book was never published, but preliminary notes are available from the Department of Computer Science at the University of Chicago, or elsewhere on the internet.
One last comment: In some of the (future) courses you have mentioned, the linear algebra one encounters takes place on infinite dimensional spaces, where the theory also requires ideas of topology and continuity. The appropriate setting for these topics is a course on functional analysis.
My suggestions for more theoretical linear algebra texts are:
- Hoffman and Kunze
- Shilov (Dover, so more "classical" but still useful)
- Lang
Since you've had some form of real analysis, definitely start reading about functional analysis. I recommend Kreyszig or "Intro to Hilbert Space" by Young, or if you're feeling more brave, Reed & Simon, Lax, or Rudin.
Also, I noticed that my understanding of linear algebra went way up by studying a bit of abstract algebra, so maybe check out a book on groups too. Matrix groups are an added bonus, because they tackle linear algebra and analysis (maybe look into this set of lecture notes).
Also also, never underestimate the power of numerical linear algebra as a way to motivate and learn the theory. Even if you're not in to numerical methods, you'll never learn quite so much as when you have to implement it for real. Trefethen and Bau is a standard text there.
There are some fundamental results that come up in both finite dimensional LA and functional analysis. The mathematical physics texts like Reed and Simon or Courant and Hilbert usually do a good job of emphasizing this. Some of the key ones are:
Less emphasized in finite dimensional linear algebra but very important is functional calculus.
Best Answer
If Linear Algebra Done Right doesn't work, then try Linear Algebra Done Wrong, by Sergei Treil. This seems to meet both of your requirements.