Linear Algebra

One of our goals in this book is to equip the reader with a unifying view of linear algebra, or at least of what is traditionally studied under this name in university courses. With this mission in mind, we start with a preview of the subject, and describe its main achievements in lay terms. To begin with a few words of praise: linear algebra is a very simple and useful subject, underlying most of other areas of mathematics, as well as its applications to physics, computer science, engineering, and economics. What makes linear algebra useful and efficient is that it provides ultimate solutions to several important mathematical problems. Furthermore, as should be expected of a truly fruitful mathematical theory, the problems it solves can be formulated in a rather elementary language, and make sense even before any advanced machinery is developed. Even better, the answers to these problems can also be described in elementary terms (in contrast with the justification of those answers, which better be postponed until adequate tools are developed). Finally, those several problems we are talking about are similar in their nature; namely, they all have the form of problems of classification of very basic mathematical objects. Yet unready to discuss the general idea of classification in mathematics, we start off with a geometric introduction to vectors, and a summary of compex numbers. Then we work out a non-trivial model example: classification of quadratic curves on the plane. Then, with this example in mind, we will be able to say a few general words about the idea of classification in general, and then present in elementary, down-to-earth terms the main problems of linear algebra, and the answers to these problems. At that point, the layout of the further material will also become clear.