Linear Algebra: vector spaces, matrices and applications

Acquiring a solid command of linear algebra requires, first and foremost, an understanding of the structures that form its foundation. This volume aims to develop such an understanding through an exposition that combines theoretical rigor with attention to computational and operational aspects. Each chapter includes exercises, which are fully solved in a dedicated section at the end of the book.

The treatment develops in an organic manner the entire program of a university course in linear algebra. We begin with the operational foundations: systems of linear equations, Gaussian elimination, matrix calculus, and the theory of the determinant. We then build the conceptual heart of the subject: the axiomatic definition of vector space, the Steinitz theorem, the notion of basis and dimension, and the Grassmann formula. Linear maps translate these structures into the language of associated matrices and change of basis. The subsequent chapters address eigenvalues, diagonalization, and their applications, powers of matrices, recurrence sequences, and differential systems, and then develop the theory of the inner product, Euclidean spaces, the Gram-Schmidt process, and the spectral theorem for real symmetric matrices. The volume closes with bilinear and quadratic forms, the classification of conics and quadrics, and the Jordan canonical form together with the matrix exponential.

The path is structured in a sequential and progressive way: every chapter uses the material of the previous ones and introduces new concepts gradually. Every important result is proved; every definition is followed by worked examples and, where possible, by a figure that makes the geometric meaning visible. The observations scattered throughout the text point out the most frequent errors and the subtleties that are easy to overlook on a first reading. The exercises at the end of each chapter allow the reader to consolidate what has been learned and to verify understanding. The book presupposes only the mathematics of the first year of university, sets, functions, elementary trigonometry, and does not require any prerequisite in linear algebra. It is intended for students of Mathematics, Physics, Engineering, and Computer Science who wish to build a solid competence in the subject.