Purpose, Gap, and Challenge
The purpose of this text is to present the contemporary theory and applications of linear algebra
to university students studying mathematics, engineering, or applied science at the postcalculus level.
Because linear algebra is usually encountered between basic problem solving courses such
as calculus or differential equations and more advanced courses that require students to cope with
mathematical rigors, the challenge in teaching applied linear algebra is to expose some of
the
scaffolding
while conditioning students to appreciate the utility and beauty of the subject.
Effectively meeting this challenge and bridging the inherent gaps between
basic and more advanced mathematics are primary goals of this book.
Rigor and Formalism
To reveal portions of the
scaffolding,
narratives, examples, and
summaries are used in place of the formal definition-theorem-proof development. But while
well-chosen examples can be more effective in promoting understanding than rigorous proofs, and
while precious classroom minutes cannot be squandered on theoretical details, I believe that
all scientifically oriented students should be exposed to some degree of mathematical thought,
logic, and rigor. And if logic and rigor are to reside anywhere, they have to be in the textbook.
So even when logic and rigor are not the primary thrust, they are always available. Formal
definition-theorem-proof designations are not used, but definitions, theorems, and proofs
nevertheless exist, and they become evident as a student's maturity increases. A
significant effort is made to present a linear development that avoids forward references,
circular arguments, and dependence on prior knowledge of the subject.
This results in some inefficiencies --- e.g., the matrix 2-norm is presented before
eigenvalues or singular values are thoroughly discussed. To compensate, I
try to provide enough ``wiggle room'' so that an instructor can temper the inefficiencies
by tailoring the approach to the students' prior background.
Comprehensiveness and Flexibility
A comprehensive treatment of linear algebra and its applications is presented and, consequently,
the book is not meant to be devoured cover-to-cover in a typical one-semester
course. However, the presentation is structured to provide flexibility in topic
selection so that the text can be easily adapted to meet the demands of different course outlines without
suffering breaks in continuity.
Each section contains basic material paired with
straightforward explanations, examples, and exercises. But every section also contains a degree of
depth coupled with thought-provoking examples and exercises that can take interested students to
a higher level.
The exercises
are formulated not only to make a
student think about material from a current section, but they are designed also to pave the way for
ideas in future sections in a smooth and often transparent manner.
The text accommodates a variety of
presentation levels by allowing instructors to select sections, discussions, examples, and exercises of
appropriate sophistication.
For example, traditional one-semester undergraduate courses
can be taught from the basic material.
An upper-level course might consist of a quick review of basics followed
by a more in-depth treatment of topics appearing later in the text.
A rich two-semester course can be taught by using the text in its entirety.
What Does ``Applied'' Mean?
Most people agree that linear algebra is at the heart of applied science, but there are divergent
views concerning what ``applied linear algebra'' really means; the academician's perspective is not
always the same as that of the practitioner.
In a poll conducted by SIAM in preparation for one
of the triannual SIAM conferences on applied linear algebra, a diverse group of internationally
recognized scientific corporations and government laboratories was asked how linear algebra finds
application in their missions. The overwhelming response was that the primary use of linear algebra
in applied industrial and laboratory work involves the development, analysis, and implementation of
numerical algorithms along with some discrete and statistical modeling. The applications in this
book tend to reflect this realization.
While most of the popular ``academic'' applications are included, and ``applications'' to other
areas of mathematics are honestly treated, there is an emphasis on
numerical issues designed to prepare students to use linear algebra in
scientific environments outside the classroom.
Computing Projects
Computing projects help solidify concepts,
and I include many exercises that can be incorporated
into a laboratory setting. But my goal is to write a
mathematics text that can last, so I don't
muddy the development by marrying the material to a
particular computer package or language. I am old enough
to remember what happened to the FORTRAN- and APL-based calculus and
linear algebra texts that came to market in the 1970s.
I provide instructors with a flexible environment that
allows for an ancillary computing laboratory in
which any number of popular packages and lab manuals can be used
in conjunction with the material in the text.
History
Finally, I believe that revealing only the
scaffolding
without
teaching something about the scientific architects who erected it deprives students of an important part of their
mathematical heritage. It also tends to dehumanize
mathematics, which is the epitome of
human endeavor. Consequently, I make an effort to say things (sometimes very human things that are not always complimentary)
about the lives of the people who contributed to the development and
applications of linear algebra.
But, as I came to realize, this is a perilous task because writing history is frequently
an interpretation of facts rather than a statement of facts.
I considered documenting the sources of the historical remarks to help mitigate the inevitable challenges,
but it soon became apparent that the sheer volume required to do so would skew the direction and flavor of the text. I can only assure
the reader that I made an effort to be as honest as possible, and I tried to corroborate ``facts.'' Nevertheless, there were times when
interpretations had to be made, and these were no doubt influenced by my own views and experiences.