Polar Sun

QuotePreface

This book helps students to master the material of a standard US undergraduate first course in Linear Algebra.

The material is standard in that the subjects covered are Gaussian reduction, vector spaces, linear maps, determinants, and eigenvalues and eigenvectors. Another standard is book's audience: sophomores or juniors, usually with a background of at least one semester of calculus. The help that it gives to students comes from taking a developmental approach — this book's presentation emphasizes motivation and naturalness, using many examples as well as extensive and careful exercises.

The developmental approach is what most recommends this book so I will elaborate. Courses at the beginning of a mathematics program focus less on theory and more on calculating. Later courses ask for mathematical maturity: the ability to follow different types of arguments, a familiarity with the themes that underlie many mathematical investigations such as elementary set and function facts, and a capacity for some independent reading and thinking. Some programs have a separate course devoted to developing maturity and some do not. In either case, a Linear Algebra course is an ideal spot to work on this transition. It comes early in a program so that progress made here pays off later but also comes late enough that students are serious about mathematics. The material is accessible, coherent, and elegant. There are a variety of argument styles, including direct proofs, proofs by contradiction, and proofs by induction. And, examples are plentiful.

Helping readers start the transition to being serious students of mathematics requires taking the mathematics seriously so all of the results here are proved. On the other hand, we cannot assume that students have already arrived and so in contrast with more advanced texts this book is filled with examples, often quite detailed.

Some books that assume a not-yet-sophisticated reader begin with extensive computations of linear systems, matrix multiplications, and determinants. Then, when vector spaces and linear maps finally appear and definitions and proofs start, the abrupt change can bring students to an abrupt stop. While this book begins with linear reduction, from the start we do more than compute. The first chapter includes proofs showing that linear reduction gives a correct and complete solution set. Then, with the linear systems work as motivation so that the study of linear combinations is natural, the second chapter starts with the definition of a real vector space. In the schedule below this happens at the start of the third week.

Another example of this book's emphasis on motivation and naturalness is that the third chapter on linear maps does not begin with the definition of homomorphism. Instead it begins with the definition of isomorphism, which is natural: students themselves observe that some spaces are "the same" as others. After that, the next section takes the reasonable step of isolating the operationpreservation idea to define homomorphism. This approach loses mathematical slickness but it is a good trade because it gives to students a large gain in sensibility.

A student progresses most in mathematics while doing exercises. In this book problem sets start with simple checks and range up to reasonably involved proofs. Since instructors usually assign about a dozen exercises I have tried to put two dozen in each set, thereby giving a selection. There are even a few that are puzzles taken from various journals, competitions, or problems collections. These are marked with a '?' and as part of the fun I have retained the original wording as much as possible.

That is, as with the rest of the book the exercises are aimed to both build an ability at, and help students experience the pleasure of, doing mathematics. Students should see how the ideas arise and should be able to picture themselves doing the same type of work.

Applications and computers. The point of view taken here, that students should think of Linear Algebra as about vector spaces and linear maps, is not taken to the complete exclusion of others. Applications and computing are interesting and vital aspects of the subject. Consequently each of this book's chapters closes with a few topics in those areas. They are brief enough that an instructor can do one in a day's class or can assign them as independent or small-group projects. Most simply give a reader a taste of the subject, discuss how Linear Algebra comes in, point to some further reading, and give a few exercises. Whether they figure formally in a course or not these help readers see for themselves that Linear Algebra is a tool that a professional must master.