Page 5 - Matrix Analysis & Applied Linear Algebra

Preface

Scaffolding
Reacting to criticism concerning the lack of motivation in his writings,
Gauss remarked that architects of great cathedrals do not obscure the beauty
of their work by leaving the scaffolding in place after the construction has been
completed. His philosophy epitomized the formal presentation and teaching of
mathematics throughout the nineteenth and twentieth centuries, and it is still
commonly found in mid-to-upper-level mathematics textbooks. The inherent ef-
ficiency and natural beauty of mathematics are compromised by straying too far
from Gauss’s viewpoint. But, as with most things in life, appreciation is gen-
erally preceded by some understanding seasoned with a bit of maturity, and in
mathematics this comes from seeing some of the scaffolding.

Purpose, Gap, and Challenge
The purpose of this text is to present the contemporary theory and applica-
tions of linear algebra to university students studying mathematics, engineering,
or applied science at the postcalculus level. Because linear algebra is usually en-
countered between basic problem solving courses such as calculus or differential
equations and more advanced courses that require students to cope with mathe-
matical 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 matrix2-norm is presented