Linear algebra is one of the most important branches of mathematics - important because of its many applications to other areas of mathematics, and important because it contains a wealth of ideas and results which are basic to pure mathematics. This book gives an introduction to linear algebra, and develops and proves its fundamental properties and theorems taking a pure mathematical approach. A large number of examples, exercises and problems are provided. Answers and/or sketch solutions to all of the problems are given in an appendix. The intended readership is undergraduate mathematicians, also anyone who requires a more than basic understanding of the subject. This book will be most useful for a "second course" in linear algebra, that is for students that have seen some elementary matrix algebra. But as all terms are defined from scratch, the book can be used for a "first course" for more advanced students.