Page 12 - A Course in Linear Algebra with Applications

Preface xi

finding optimal solutions, which texts at this level often fail
to cover.
Chapter Eight introduces the reader to the theory of
eigenvectors and eigenvalues, still one of the most powerful
tools in linear algebra. Included is a detailed account of ap-
plications to systems of linear differential equations and linear
recurrences, and also to Markov processes. Here we have not
shied away from the more difficult case where the eigenvalues
of the coefficient matrix are not all different.
The final chapter contains a selection of more advanced
topics in linear algebra, including the crucial Spectral Theo-
rem on the diagonalizability of real symmetric matrices. The
usual applications of this result to quadratic forms, conies
and quadrics, and maxima and minima of functions of several
variables follow.
Also included in Chapter Nine are treatments of bilinear
forms and Jordan Normal Form, topics that are often not con-
sidered in texts at this level, but which should be more widely
known. In particular, canonical forms for both symmetric and
skew-symmetric bilinear forms are obtained. Finally, Jordan
Normal Form is presented by an accessible approach that re-
quires only an elementary knowledge of vector spaces.
Chapters One to Eight, together with Sections 9.1 and
9.2, correspond approximately to a one semester course taught
by the author over a period of many years. As time allows,
other topics from Chapter Nine may be included. In practice
some of the contents of Chapters One and Two will already be
familiar to many readers and can be treated as review. Full
proofs are almost always included: no doubt some instructors
may not wish to cover all of them, but it is stressed that for
maximum understanding of the material as many proofs as
possible should be read. A good supply of problems appears
at the end of each section. As always in mathematics, it is an