Linear Algebra

Abstract

This is a translation of the (slightly revised) second German edition of our book “Lineare Algebra”, published by Springer Spektrum in 2015. Our general view of the field of Linear Algebra and the approach to it that we have chosen in this book were already described in our Preface to the First German Edition, published by Vieweg+Teubner in 2012. In a nutshell, our exposition is matrix-oriented, and we aim at presenting a rather complete theory (including all details and proofs), while keeping an eye on the applicability of the results. Many of them, though appearing very theoretical at first sight, are of an immediate practical relevance. In our experience, the matrix-oriented approach to Linear Algebra leads to a better intuition and a deeper understanding of the abstract concepts, and therefore sim- plifies their use in real-world applications. Starting from basic mathematical concepts and algebraic structures we develop the classical theory of matrices, vectors spaces, and linear maps, culminating in the proof of the Jordan canonical form. In addition to the characterization of important special classes of matrices or endomorphisms, the last chapters of the book are devoted to special topics: Matrix functions and systems of differential equations, the singular value decomposition, the Kronecker product, and linear matrix equations. These chapters can be used as starting points of more advanced courses or seminars in Applied Linear Algebra.