This book covers all the fundamental topics of linear algebra and outlines some of its applications. Unlike the majority of mathematical textbooks, it does not use the standard theorem-proof approach. The authors explain linear algebra by means of examples and its practical and geometric applications. The reader is led to an intuitive understanding of the notions of the subject and provided with a nice survey of their applications. The basic topics (vectors, linear maps, linear systems, affine maps, and eigenvalues and eigenvectors) are explained first in two dimensions. Then the same concepts are retained and extended in a three dimensional setting, and finally general linear systems of equations and general vector spaces are defined and studied. In addition, several chapters are devoted to geometric and practical applications including the analysis of conics, polygons, triangulations, numerical methods, and curves. The book also contains a brief postscript tutorial chapter and solutions to selected problems. The book is designed for students of fields using linear algebra, such as engineering or computer science. The text could help specialists in these branches to understand the mathematical background of the methods they use. It is also an inspiring book for pure mathematicians who would like to learn more about applications of linear algebra. It is definitely a useful source of many nice examples and applications for teachers of linear algebra.