This book is an introduction to the concepts of Linear Algebra and Analytic Geometry. It is based on the lectures I have given in the last five years to first-year students of the Department of Mechanical Engineering at the Polytechnic School of the Aristotle University of Thessaloniki. The book is mainly aimed at students of Engineering Schools and Schools of Natural Sciences. Its purpose is to present the basic concepts, avoiding, for example, the proofs of theorems and emphasizing their applications. For this reason, a plethora of solved examples as well as unsolved exercises have been included in each chapter. [...]