These notes are an expanded version of the seven hours of lectures I gave at Lancaster. I have kept to the original plan and policy, which perhaps need some explanation. Roughly speaking, the contents are what I should like my own graduate students to know about Lie groups, and my general idea was to show how the theory is a natural continuation of basic linear algebra. As root systems and the classification of semisimple Lie algebras were treated in the companion lecture courses I felt I had an excuse for concentrating firmly on the general linear groups. But in any case I believe that is the right way to approach the subject: the taxonomic side of the theory is not to my taste.

I tried to make my lectures useful to people with rather different amounts of mathematical knowledge and sophistication. That means the level is uneven: remarks aimed at the more advanced readers are scattered throughout, and are meant to be ignored by others. I hope the chapters can be read in almost any order: I tried to make them fairly independent. The first four are devoted to a survey of concrete examples of the theory to be developed. This is mainly “undergraduate” material, and so I put it before the formal definition of a Lie group in Chapter 5. But it does not need to be read in advance, and sometimes it uses terminology which is defined only later.