This book originates from the lessons held by the author in university courses and is aimed at students who, for the first time, are approaching a course in linear algebra and geometry. Bearing in mind the difficulties that students usually encounter in the study of abstract topics such as those presented in this book, we have chosen to use a language that is as simple as possible, trying to motivate the introduction of the various abstract notions with concrete examples. Topics covered include the theory of vector spaces and linear functions, the theory of matrices and systems of linear equations, the theory of Euclidean vector spaces and, finally, the applications of linear algebra to the study of the geometry of affine space. Numerous figures, examples and exercises carried out in every detail have been included in order to facilitate the study and understanding of the topics presented.