Linear Algebra

Description

This self-contained textbook takes a matrix-oriented approach to linear algebra and presents a complete theory, including all details and proofs, culminating in the Jordan canonical form and its proof. Throughout the development, the applicability of the results is highlighted. Additionally, the book presents special topics from applied linear algebra including matrix functions, the singular value decomposition, the Kronecker product and linear matrix equations. The matrix-oriented approach to linear algebra leads to a better intuition and a deeper understanding of the abstract concepts, and therefore simplifies their use in real world applications. Some of these applications are presented in detailed examples. In several ‘MATLAB-Minutes’ students can comprehend the concepts and results using computational experiments. Necessary basics for the use of MATLAB are presented in a short introduction. Students can also actively work with the material and practice their mathematical skills in more than 300 exercises. Front Matter....Pages i-xi Linear Algebra in Every Day Life....Pages 1-7 Basic Mathematical Concepts....Pages 9-21 Algebraic Structures....Pages 23-35 Matrices....Pages 37-53 The Echelon Form and the Rank of Matrices....Pages 55-71 Linear Systems of Equations....Pages 73-79 Determinants of Matrices....Pages 81-99 The Characteristic Polynomial and Eigenvalues of Matrices....Pages 101-113 Vector Spaces....Pages 115-133 Linear Maps....Pages 135-154 Linear Forms and Bilinear Forms....Pages 155-166 Euclidean and Unitary Vector Spaces....Pages 167-186 Adjoints of Linear Maps....Pages 187-197 Eigenvalues of Endomorphisms....Pages 199-212 Polynomials and the Fundamental Theorem of Algebra....Pages 213-226 Cyclic Subspaces, Duality and the Jordan Canonical Form....Pages 227-251 Matrix Functions and Systems of Differential Equations....Pages 253-269 Special Classes of Endomorphisms....Pages 271-293 The Singular Value Decomposition....Pages 295-302 The Kronecker Product and Linear Matrix Equations....Pages 303-310 Back Matter....Pages 311-324