Linear Algebra, Theory And Applications, 2012a

Preface
This is a book on linear algebra and matrix theory. While it is self contained, it will work
best for those who have already had some exposure to linear algebra. It is also assumed that
the reader has had calculus. Some optional topics require more analysis than this, however.
I think that the subject of linear algebra is likely the most significant topic discussed in
undergraduate mathematics courses. Part of the reason for this is its usefulness in unifying
so many different topics. Linear algebra is essential in analysis, applied math, and even in
theoretical mathematics. This is the point of view of this book, more than a presentation
of linear algebra for its own sake. This is why there are numerous applications, some fairly
unusual.
This book features an ugly, elementary, and complete treatment of determinants early
in the book. Thus it might be considered as Linear algebra done wrong. I have done this
because of the usefulness of determinants. However, all major topics are also presented in
an alternative manner which is independent of determinants.
The book has an introduction to various numerical methods used in linear algebra.
This is done because of the interesting nature of these methods. The presentation here
emphasizes the reasons why they work. It does not discuss many important numerical
considerations necessary to use the methods effectively. These considerations are found in
numerical analysis texts.
In the exercises, you may occasionally see ↑ at the beginning. This means you ought to
have a look at the exercise above it. Some exercises develop a topic sequentially. There are
also a few exercises which appear more than once in the book. I have done this deliberately
because I think that these illustrate exceptionally important topics and because some people
don’t read the whole book from start to finish but instead jump in to the middle somewhere.
There is one on a theorem of Sylvester which appears no fewer than 3 times. Then it is also
proved in the text. There are multiple proofs of the Cayley Hamilton theorem, some in the
exercises. Some exercises also are included for the sake of emphasizing something which has
been done in the preceding chapter.
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