Reg Allenby - Linear Algebra

This series is designed particularly, but not exclusively, for students reading degree programmes based on semester-long modules. Each text will cover the essential core of an area of mathematics and lay the foundation for further study in that area. Some texts may include more material than can be comfortably covered in a single module, the intention there being that the topics to be studied can be selected to meet the needs of the student. Historical contexts, real life situations, and linkages with other areas of mathematics and more advanced topics are included. Traditional worked examples and exercises are augmented by more open-ended exercises and tutorial problems suitable for group work or self-study. Where appropriate, the use of computer packages is encouraged. The first level texts assume only the A-level core curriculum.

Chris D. Collinson, Professor

Johnston Anderson, Dr

Peter Holmes, Mr

This book is, somewhat loosely, based on a (module length) series of lectures which have been given, over the years, to the first year honours mathematics students in the University of Leeds. The looseness derives from the fact that I have included here rather more material than I could have got through in the 24 lectures at my disposal. In particular I had no time for the applications nor the historical and biographical details I have included here nor, in fact, for all parts of the main text This is chiefly because I have expanded and rearranged my original notes substantially to take account of the fact that, as distinct from the lecturer, the writer is not available to answer questions.

The point of including the applications is to inform those genuinely interested (and, perhaps more importantly, those who, being less readily convinced, ask or should ask O.K., but whats all this stuff good for?) of why there is merit in studying linear algebra. In fact the biographies also help in this regard since they sometimes indicate from what problem(s) an idea or whole theme emerged.

. Most are pretty trivial in content (after all I had to be able to do them!) and my main hope is that the reader will invent some more interesting (or outrageous) ones for him or herself. Incidentally I used the Maple package mainly because it is on Leeds computer system and, Im told, likely to be available to widespread audiences for some time.

A word about the chapter contents. The first five chapters are fairly computational in nature and, because they deal with ideas (systems of simultaneous linear equations, matrices, their inverses and their determinants) which will be familiar to a good many readers, they should be reasonably straightforward. So, to show that one can have slightly deeper thoughts even about straightforward material and, in particular, to encourage YOU to have such thoughts the text includes what I hope are thought provoking Tutorial Problems which those readers in higher education can discuss with their tutors! Such problems persist throughout the book, there being some 28 in total.

some time in his or her undergraduate career and, with so many almost identical examples to hand, it seems like too good an opportunity to miss.

(which is quite computational) then investigates how one might simplify these matrices for the purpose of aiding calculation.

At a quick glance the book might appear to move slowly from being very computational to being more words than numbers. I personally think it is more a case of moving from the familiar to the newer. Nevertheless, even the more wordy bits are backed up by numerous examples. Furthermore, apart from the almost 350 exercises to be found at the ends of chapters, there are over 60 Problems given as the subject is developed, the purpose of each being to reinforce what has just been described in the text.

Finally it is my great pleasure to drank a number of people who have helped me to produce this book. These include Dr Jeremy Gray of the Open University for his assistance, willingly given, relating to the biographies, Professor Dr Konrad Jacobs who (as with my other books) has generously supplied the photographs and my colleague Dr Eric Wallace who learnt Maple to test out my computer package problems (and sometimes found me wanting!)

Last, but not least, I should like to thank Dr Johnston Anderson both for inviting me to write this book and for reminding me that explanations which can readily be given verbally in lectures do need to be written (and sometimes at greater length) in a book. In addition, his suggestions concerning the elimination of a number of mathematical jokes were, in hindsight, correct. Needless to say he takes no responsibility for the quality of those remaining.

Finally, in these days when we are required to be assessed on every aspect of our lives, I really do implore any reader (especially of the undergraduate kind) who has other than normal feelings of pleasure or annoyance in what he or she finds (or doesnt find) in this book to let me know of these feelings so that I may write a better book next time round. And if any reader should know of any really good mathematical joke(s) especially ones concerning linear algebra.

RBJTA, Leeds 1994

Solutions of systems of simultaneous linear equations (also called linear systems) arise in very many real life situations; for just a couple of instances see the Applications at the end of the chapter. Here we show how to solve such systems (much as a computer would) by simplifying them in a systematic way. We shall also see how to interpret the results we obtain geometrically.

In 1849 the French mathematician Joseph Alfred Serret (30 August 18192 March 1885) wrote: Algebra is, properly speaking, the analysis of equations. (This is no longer an accurate statement although the motivating factor behind the theory of groups was the investigation of the solutions of polynomial equations.) Accordingly,