This small booklet published in the series Student Mathematical Library of the AMS is devoted to a basic course on linear algebra as taught in standard undergraduate courses on the topic. It covers (in an economical way) all the necessary topics to be expected (vector spaces, basis and dimension, systems of linear equations, linear maps and their relations to matrices, determinants, linear functionals and adjoint maps, inner-product spaces with a discussion of normal, unitary and orthogonal operators, nilpotent operators and the Jordan canonical form, and characteristic and minimal polynomials of a map). The last chapter also covers the basics of quadratic forms, the Perron-Frobenius theory, stochastic matrices and representations of finite groups in a reduced form. The appendix contains a review of the necessary prerequisites (in particular concerning polynomials). The book is written in an elegant, condensed way. It contains many exercises, mostly of theoretical character. The main advantage (in particular for teachers and talented students) is that basic ideas are carefully isolated and presented in a simple, minimal and understandable way. It is a very good complement to many other books containing calculus, specific examples, and applications of linear algebra.