This book originates from a collection of lecture notes that the first author prepared at the University of Trieste with Michela Brundu, over a span of fifteen years, together with the more recent one written by the second author. The notes were meant for undergraduate classes on linear algebra, geometry and more generally basic mathematical physics delivered to physics and engineering students, as well as mathematics students in Italy, Germany and Luxembourg.
The book is mainly intended to be a self-contained introduction to the theory of finite-dimensional vector spaces and linear transformations (matrices) with their spectral analysis both on Euclidean and Hermitian spaces, to affine Euclidean geometry as well as to quadratic forms and conic sections.
Many topics are introduced and motivated by examples, mostly from physics. They show how a definition is natural and how the main theorems and results are first of all plausible before a proof is given. Following this approach, the book presents a number of examples and exercises, which are meant as a central part in the development of the theory. They are all completely solved and intended both to guide the student to appreciate the relevant formal structures and to give in several cases a proof and a discussion, within a geometric formalism, of results from physics, notably from mechanics (including celestial) and electromagnetism.