This well-established and continuously revised textbook, proven over 50 years, is ideally suited as a foundation for a two-semester introductory lecture for students of mathematics, physics, and computer science, as well as for other subjects that require mathematical fundamentals from linear algebra. Some advanced topics can be easily skipped for a quick start. Throughout the text, abstract concepts are motivated by examples, and the lively interrelationships between general theory and concrete calculations using matrices are emphasized. The text contains numerous exercises, many solutions to which can be found in the exercise book on linear algebra authored by H. Stoppel and B. Griese. Additional topics and applications are covered in the study book "Linear Algebra and Analytical Geometry" by Gerd Fischer, which is an excellent supplement for self-study. In the 20th edition, alongside several minor improvements, particularly in Chapter 7, the sections on tensor products and multilinear algebra have been revised and expanded, as these parts typically pose special difficulties during the first reading. However, they are important for understanding the algebraic background in integration theory.