This book contains basics of linear algebra (vector spaces, subspaces, bases, linear independence and span, lines and places, solving linear systems, linear transformations and its properties, matrices and operations with them, matrix representations of transformations, determinants and their properties, and eigenvalues and eigenvectors) and it focuses on students starting their studies of linear algebra with almost no experience reading and studying mathematical texts. To encourage them, the book has a specific structure and it does not use the compressed language typical of other mathematical textbooks. Each chapter starts with a short discussion of motivation and solving strategy. Then solutions of some special methodical exercises are given with many explanatory comments. The chapter continues with a typical mathematical text (definition, proposition and proof) and it finishes with various applications and exercises. The author explains step-by-step many solutions of some exercises and integrates a lot of examples of real applications into the theoretical text. He also offers simple instructions for using Maple, MATLAB and TI-83 Plus to give the students the keystrokes to reduce the manual labour associated with many types of exercises, as well as to allow them to focus on the problem at hand and to do more exercises in the same amount of time. The book can be recommended to students beginning their first mathematical course at university, as well as to teachers who want to improve their style of teaching or who are looking for different ways of presenting the basics of linear algebra.