This will be one of the most important classes you ever take. Linear algebra and calculus are the foundations of modern mathematics and its applications; the language and viewpoint of linear algebra is so thoroughly woven into the fabric of mathematical reasoning that experienced mathematicians, scientists, and engineers can forget it is there, in the same way that native speakers of a language seldom think consciously about its formal structure. Achieving this fluency is a big part of that nebulous goal of “mathematical maturity.”
In the context of your mathematical education, this book marks an important transition. In it, you will move away from a largely algorithmic, problem-centered viewpoint toward a perspective more consciously grounded in rigorous theoretical mathematics. Making this transition is not easy or immediate, but the rewards of learning to think like a mathematician run deep, no matter what your ultimate career goals are. With that in mind, we wrote this book to be read – by you, the student. Reading and learning from an advanced mathematics text book is a skill, and one that we hope this book will help you develop.
There are some specific features of this book aimed at helping you get the most out of it. Throughout the book, you will find “Quick Exercises,” whose answers are usually found (upside down) at the bottom of the page. These are exercises which you should be able to do fairly easily, but for which you may need to write a few lines on the back of an envelope. They are meant to serve as checkpoints; do them! The end of each section lists “Key Ideas,” summarizing (sometimes slightly informally) the big picture of the section. Certain especially important concepts on which there are many important perspectives are summarized in features called “Perspectives” at the end of some chapters. There is an appendix covering the basics of sets, functions, and complex number arithmetic, together with some formal logic and proof techniques. And of course, there are many exercises. Mathematics isn't something to know, it's something to do; it is through the exercises that you really learn how.