These volumes introduce and develop the theory of linear structures for students of mathematics and its applications. Today, linear algebra serves as an essential tool and unifying language across nearly all areas of mathematics. Given its importance in the natural sciences, engineering, and economics, linear algebra is presented as a valuable and widely applicable subject in its own right.

Adopting an inductive approach while maintaining an appropriate level of abstraction, the volumes lead the reader from finite-dimensional vector spaces over the fields ℝ and ℂ to general fields and infinite-dimensional vector spaces. Throughout, the principal examples are the tuple space ℝⁿ, matrix spaces, and solution spaces of systems of linear equations.

To preserve accessibility, the exposition does not follow a strictly deductive development. Instead, inductive elements are incorporated throughout. In the first chapter, margin notes indicate where the theory is developed through the study of systems of linear equations.  Further margin notes indicate where the general theory is applied to systems of linear equations. Different methods of proof are used to illuminate the same topic from complementary perspectives.

A recurring theme is the algorithmic viewpoint, emphasizing practical methods and numerical complexity without sacrificing structural understanding. The discussion therefore begins with the theoretical insights provided by Gaussian elimination and develops the corresponding structural concepts alongside computational techniques. 

In addition to the standard core material, the volume includes topics of interest to students from a variety of disciplines. Students of mathematics education will find an introduction to several aspects of analytic geometry, beginning in this first volume with affine geometry. Readers drawn to algebra are introduced not only to the theory of vector spaces over a field K, but also to more general algebraic structures. For students pursuing numerical mathematics, optimization, or applications outside mathematics, topics such as LU decomposition, least-squares problems and the pseudoinverse are discussed. For students of physics a series of examples incorporates the linear algebra related aspects of a first year physics course.

Particular emphasis is placed on connecting theory and algorithms and on relating both to applications in the sciences. To support this goal, mathematical modelling plays a central role. Ongoing examples drawn from mechanics, electrical networks, and economics are developed progressively alongside the theory. An additional running example explores historical questions.

Peter Knabner is a German mathematician and currently Professor Emeritus of Applied Mathematics at the University Erlangen-Nuremberg after over 20 years in office as chaired professor. In his research he developed and analyzed mathematical models and their numerical schemes for multiphase flow and reactive multicomponent transport in porous media. He is the author of well over 160 peer-reviewed publications in applied analysis, numerical mathematics, and hydrogeology, as well as author and co-author of 14 monographs and textbooks in English and German, including those on the numerical methods for partial differential equations, mathematical modeling, and linear algebra.

Wolf Barth was a German mathematician and has been in office as a chaired professor at the University Erlangen-Nuremberg for over 30 years. In his research, he investigated vector bundles and projective surfaces (Barth surface). He had a strong dedication to teaching, in particular linear algebra.