PrefaceThis is a book on linear algebra and matrix theory. While it is self contained, it will workbest for those who have already had some exposure to linear algebra. It is also assumed thatthe 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 inundergraduate mathematics courses. Part of the reason for this is its usefulness in unifyingso many different topics. Linear algebra is essential in analysis, applied math, and even intheoretical mathematics. This is the point of view of this book, more than a presentationof linear algebra for its own sake. This is why there are numerous applications, some fairlyunusual.

This book features an ugly, elementary, and complete treatment of determinants earlyin the book. Thus it might be considered as Linear algebra done wrong. I have done thisbecause of the usefulness of determinants. However, all major topics are also presented inan 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 hereemphasizes the reasons why they work. It does not discuss many important numericalconsiderations necessary to use the methods effectively. These considerations are found innumerical analysis texts.

In the exercises, you may occasionally see ↑ at the beginning. This means you ought tohave a look at the exercise above it. Some exercises develop a topic sequentially. There arealso a few exercises which appear more than once in the book. I have done this deliberatelybecause I think that these illustrate exceptionally important topics and because some peopledon’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 alsoproved in the text. There are multiple proofs of the Cayley Hamilton theorem, some in theexercises. Some exercises also are included for the sake of emphasizing something which hasbeen done in the preceding chapter.

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