Axler's book is very good. An alternative to consider is Paul Halmos's much older Finite-Dimensional Vector Spaces [0]. Both books take the same basic approach, and the proofs of the major theorems are substantially the same. (Halmos's book is well known and well liked and was probably Axler's starting point.)
Axler's book covers more ground (most notably, Halmos presents the polar decomposition but not the singular-value decompostion) and uses more modern terminology and notation. But Halmos's book has the merits of being half as long and a third as expensive, as well as having been written specifically to prepare the reader as directly as possible for Halmos's short introduction to Hilbert spaces [1].
I highly recommend one or the other of these books for readers who want to understand linear algebra as mathematicians do.
For those who already know enough linear algebra to be bored with an entire book dedicated to Axler's approach to linear algebra, you can simply read the paper upon which the book is based:
Axler's book covers more ground (most notably, Halmos presents the polar decomposition but not the singular-value decompostion) and uses more modern terminology and notation. But Halmos's book has the merits of being half as long and a third as expensive, as well as having been written specifically to prepare the reader as directly as possible for Halmos's short introduction to Hilbert spaces [1].
I highly recommend one or the other of these books for readers who want to understand linear algebra as mathematicians do.
0. https://www.amazon.com/Finite-Dimensional-Vector-Spaces-Paul...
1. https://www.amazon.com/Introduction-Hilbert-Theory-Spectral-...