Here are some questions to be answered after reading the manuscript.
* What do you think of the choice of topics?
It was not what I expected. The book gives what I would consider a
more pure than applied presentation of linear algebra. I expected more
attention to iterative methods. And I really expected to see a number
of real applications examined in some detail. The brief mentions of
applications in Chapter 14 and elsewhere in the text do not seem to be
organically woven into the book. They seem more like add-ons that are
secondary to the main themes. One does not come away from the book
feeling that you have gotten much of an idea about how linear algebra is
used in applications.
* What do you think of the authors' approach and writing style?
The writing is very precise and formal. Crystal clear, but dense. The
author does do the best job I have seen of inserting insightful comments
about the intuition mathematicians have developed concerning linear
algebra. The author has clearly worked on this manuscript for a long
time as a labor of love. An advanced student wanting precise statements
and proofs could get them from this book.
The author chose to include MatLab code as an integral part of
the text. This may be an excellent idea, but it will limit the
potential audience to a certain extent. I sympathize with its
inclusion, and do not want to suggest that it be removed.
But I do believe that since it is included, MatLab should be used to
enable the book and the readers to work with some really large scale
examples and problems. Such examples should occur right in the text.
The author has chosen to use occasionally the language of algebra,
including words such as groups, rings, Hilbert basis theorem, that he
reviews in the background chapter. While this is not a big deal, it
will make the book seem a little more formidable than it is to many
potential readers. Particularly the most applied students may feel left
out.
* What sort of audience do you think this book will find?
I can not imagine using this as a textbook for a course. It could be a
reference for an advanced linear algebra course. Realistically, I think
that only graduate students in mathematics will be able to appreciate
it. A beginning graduate student trying to solidify his understanding
of linear algebra might well consult this book (along with others) quite
frequently, with profit.
I can not imagine, for example, engineering students, even engineering
graduate students, using the book with benefit, which is what I
originally expected from the title word "Applied". The book is for
mathematicians.
* Are there topics you think the authors should consider adding
(or even omitting)?
For an applied text, I can suggest several things I would look for.
(1) Large scale problems.
(2) Some attention to iterative methods, e.g., for solving linear
systems.
(3) Some attention to implementation issues. For example, the author
mentions householder reflections as providing a better numerical scheme
for orthogonalizing than Gram-Schmidt, but it would be good to devote
some time explaining how this is done and why it is better, with
examples.
(4) Some serious topics from numerical analysis based on linear algebra.
I could well imagine a chapter on finite difference and finite element
methods for PDEs.
(5) Fast Fourier transform would be a great applied topic that is all
linear algebra. Some discrete wavelets could make the book seem more
modern.
(6) Substantial applications of more of the topics already in the book.
For example, the singular value decomposition is given but its real
power in applications is only hinted at.
* Are there similar books already published? How does this book
compare with them?
I do not know of the perfect book for APPLIED linear algebra. For a
beginning graduate course focusing on linear algebra for applications
(but not including the applications), there is Trefethen and Bau's
"Numerical Linear Algebra", that comes closer to what I would have in
mind.
At a lower level Strang's "Linear Algebra", Third edition, is great for
undergraduates. Strang's new book "Computational Science and
Engineering" takes a very applied approach to a number of topics,
including linear algebra. Strang's choice of topics is impeccable, but
his book could be hard to use.
I once taught from "Applied Linear Algebra: The Decoupling Principle" by
Lorenzo Sadun that I would prefer for a second undergraduate course in
linear algebra.
For graduate students wanting a fairly pure course, I would prefer
"Linear Algebra" by Peter Lax.
* Do you think this book should be published as it as or with
changes?
It is hard for me to know what you are looking for. I would suggest, at
the least, (1) that the applications in chapter 14 each be developed
more with real applications demonstrated; (2) that the book be sprinkled
with serious large scale examples; (3) that the author be asked for a
very clear indication of his intended audience, so that the book can be
evaluated with his goals in mind.
********************************************************************* *