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# REVIEW OF A FIRST COURSE IN LINEAR ALGEBRA

**DAVID S. WATKINS **

As textbook prices head into the stratosphere, it is
gratifying to observe that

some people are trying to do something about it. Robert Beezer has made his text
A

First Course in Linear Algebra available to the public for free. A MathML
hypertext

version is on the web at , you can download a pdf version, or

you can buy an inexpensive paper copy through the print-on-demand service
lulu.com.

The online and pdf versions contain links back to all theorems, definitions,
etc., as

they are used.

This book is a work in progress. There are three parts:
Core, Topics, and Applications,

but only the Core part resembles a finished book. The other two are just

beginning to take form. The author envisions adding more and more to the book
over

time, and he encourages all of us to help out by making contributions. To this
end he

has made the TEX source code available. Users of the book are free to modify it
and

share their modifications with others, subject to the terms of the GNU Free
Document

License. As the author notes in the preface, “[This book] will never go ‘out of
print’

nor will there ever be trivial updates designed to frustrate the used book
market.”

So what is the book like? First of all, all chapters,
sections, theorems, and other

items are given acronyms instead of numbers. Some examples: The chapter on
vector

spaces is Chapter VS. A theorem that says that linear transformations take zero
to

zero is called Theorem LTTZZ. The section on injective linear transformations is

Section ILT. This practice has the advantage that Theorem LTTZZ will always have

that name, even as the book changes shape over time. The disadvantage is that
the

text is cluttered with acronyms, most of which will have no immediate meaning
for

the reader.

The focus of the book is theoretical. Quoting again from
the preface: “This

textbook is designed to teach the university mathematics student the basics of
the

subject of linear algebra and the techniques of formal mathematics.” Thus the
book

is meant for mathematics majors, and proofs and proof techniques are emphasized.

The tone of the book is mostly formal; equations are preferred over words. Most

of the proofs contain long chains of equalities like this one taken from the
proof of

Theorem OSIS (one side is sufficient), which shows that a right inverse is also
a left

inverse:

In the electronic versions, the cited theorems have links
back to their statements. In

the paper version, page numbers are given.

This is a text for an algebra course. As the author states
in the very first section,

“. . .we will maintain an algebraic approach to the subject, with the geometry
being

secondary. . . . here and now we are laying our bias bare.” Indeed, the book is
entirely

algebraic; geometry is nowhere to be found.

The book begins concretely. There is a chapter on solving
linear systems using

row-echelon form. This is followed by a chapter on vectors in C^{n} that introduces

concepts such as linear combinations, linear independence, spanning sets, and
orthogonality.

Then a chapter on matrices introduces matrix operations, the matrix inverse,

null space, row space, column space, and the like. The relationships between
these

concepts and the solution of linear systems are laid out. After this the book
takes an

abstract turn, with chapters on vector spaces and linear transformations.
Sandwiched

between these are chapters on determinants and eigenvalues of matrices. This
ordering

strikes me as odd, but perhaps the order is not so important. The final chapter
of

the Core part of the book is on representations of linear transformations by
matrices,

culminating in the Jordan canonical form.

In my paper copy of the book, the Topics and Applications
parts are empty. On

the web there are, as of this writing, a couple of sections in Applications and
ten

or so in Topics. The author’s intent is that the Topics part will include topics
that

the author deems not to be part of the core but that some instructors might like
to

include in their courses. For example, the fact that Gaussian elimination yields
an

LU decomposition of a matrix is included in Topics and not mentioned in the
Core.

This choice reminds us once again of the bias of the book.
This is definitely a

“pure math” treatment of linear algebra. It is assumed that all arithmetic can
be done

exactly. Nowhere is it mentioned that the principal algorithms presented in the
Core

break down when executed in floating-point arithmetic on a computer or
calculator.

(For example, row reduction cannot determine reliably whether a square matrix is

singular or not.) Condition numbers are not mentioned, nor are the superior
numerical

properties of unitary transformations. The Gram-Schmidt process is introduced in
the

core, but the QR decomposition is not mentioned at all. (Hopefully it will at
least be

added as a topic later.) The singular value decomposition is included as a mere
topic.

We are told that it is useful, but no specific uses (e.g. reliable rank
determination)

are given. Of course, all of this can be changed in the future.

Here’s one other thing I would change: The index is way
too long for a single web

page. It ought to be split over many pages, perhaps 26 or so.

The preference for symbols over words sometimes has bad
consequences. For

example, at the beginning of Chapter D (determinants) a square matrix
is defined

as follows:

Got that? Here’s what I would have preferred: “
is the matrix obtained by

interchanging the ith and jth rows of the identity matrix.”

I was amused by Example CVS (crazy vector space). This is
the set of all ordered

pairs of complex numbers with the operations

(x_{1}, x_{2}) + (y_{1}, y_{2}) = (x_{1} + y_{1} + 1, x_{2} + y_{2} + 1)

α(x_{1}, x_{2}) = ( αx_{1} + α − 1, αx_{2} + α − 1).

This really is a vector space! (What is the “zero”
element?) It is a wonderful example

in the spirit of abstract algebra, but there is no way I would show it to my
sophomores.

I want to develop their intuition, not crush it.

**Would I use this book in my course?** For me the
total omission of geometry

disqualifies it immediately. A first course in linear algebra ought to contain
lots of

geometry and pictures (along with the algebra) to help those sophomores develop

some insight. Moreover, at my institution the first linear algebra course is
populated

mainly by engineering and science majors. They need to get the basics; they do
not

need an introduction to formal mathematics. That is left for other courses,
including

the second linear algebra course, which is mainly for mathematics majors.
Beezer’s

book would surely be a useful supplemental resource for that course.

Clearly Beezer and I have some philosophical differences
about what ought to be

in a first course in linear algebra. Nevertheless I commend him for undertaking
this

project. Instructors who wish to teach a pure linear algebra course that
emphasizes

rigor and formal mathematics will be able to make good use of this material and
feel

secure in the knowledge that the book is not going to go out of print. Finally,
the

price is right.