This is a bold and concise introduction to linear algebra written by a
leading applied mathematician. The book was developed, obviously over
many years, from the lecture notes for the applied linear algebra class
taught by the author. The result is unorthodox, minimalistic, and
stunning.
 
This book will be a classic and should be in every mathematical library.
However, the use and the audience of this book will likely turn out to
be very different from those naturally expected. Specifically, it will
most probably become a reference for  researchers as well as the text of
choice for  the more ambitious teachers and students of linear algebra
rather than a mainstream linear algebra textbook.  

The book challenges the reader in multiple ways. Its slick and precise
notation will take a while for anyone to master, working mathematicians
included.
 (Once mastered, it provides a powerful engine that easily 
drives most proofs.)   Many familiar concepts are presented in unusual
ways 
(e.g., bases as linear maps). ``Non-linear algebraic'' notions (e.g.,
rings and fields, compactness and continuity) and ``advanced'' topics
(e.g., Perron-Frobenius' theorem)  are all over the place. The
derivations are terse and the deciphering hints are scarce. The pace is
unforgiving of the lazy student. The book uses (some) MATLAB notation
and gives many MATLAB examples/exercises.

For these reasons, working this book out in detail will make anyone a
better mathematician. In fact, a student capable of such work without
much outside help will emerge ready to do research in mathematics.
A working mathematician will likely emerge with new insights into
his/her own area of expertise. 

For the same  reasons, the book is just as challenging for the
instructor.  
A good instructor, however, could elaborate on tricky details, work out
difficult examples, and spice up MATLAB routines to a delightful effect.

The best instructors at top universities  will seriously consider it as
a textbook for math majors. Beyond that, it may never be used for
undergraduate teaching unless the book is promoted by sufficiently many
enthusiasts.


With all that, not publishing the book would be a shame. There  are and
will continue to be written a number of mathematically solid,
slow-paced, user-friendly textbooks for standard undergraduate classes
-- the quality bread-and-butter of linear algebra. This is a work of
art.
Publish it.