The present book is based on the extensive lecture notes of the author and contains a concise course on Linear Algebra. The sections begin with an intuitive presentation, aimed at the beginners, and then often include rather non-trivial topics and exercises. This makes the book suitable for introductory as well as advanced courses on Linear Algebra.The first part of the book deals with the general idea of systems of linear equations, matrices and eigenvectors. Linear systems of differential equations are developed carefully and in great detail. The last chapter gives an overview of applications to other areas of Mathematics, like calculus and differential geometry. A large number of exercises with selected solutions make this a valuable textbook for students of the topic as well as lecturers, preparing a course on Linear Algebra.