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© 2015

Linear Algebra

Textbook

Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Table of contents

  1. Front Matter
    Pages i-xi
    PDF
  2. Linear Algebra in Every Day Life
    Jörg Liesen, Volker Mehrmann
    Pages 1-7
  3. Basic Mathematical Concepts
    Jörg Liesen, Volker Mehrmann
    Pages 9-21
  4. Algebraic Structures
    Jörg Liesen, Volker Mehrmann
    Pages 23-35
  5. Matrices
    Jörg Liesen, Volker Mehrmann
    Pages 37-53
  6. The Echelon Form and the Rank of Matrices
    Jörg Liesen, Volker Mehrmann
    Pages 55-71
  7. Linear Systems of Equations
    Jörg Liesen, Volker Mehrmann
    Pages 73-79
  8. Determinants of Matrices
    Jörg Liesen, Volker Mehrmann
    Pages 81-99
  9. The Characteristic Polynomial and Eigenvalues of Matrices
    Jörg Liesen, Volker Mehrmann
    Pages 101-113
  10. Vector Spaces
    Jörg Liesen, Volker Mehrmann
    Pages 115-133
  11. Linear Maps
    Jörg Liesen, Volker Mehrmann
    Pages 135-154
  12. Linear Forms and Bilinear Forms
    Jörg Liesen, Volker Mehrmann
    Pages 155-166
  13. Euclidean and Unitary Vector Spaces
    Jörg Liesen, Volker Mehrmann
    Pages 167-186
  14. Adjoints of Linear Maps
    Jörg Liesen, Volker Mehrmann
    Pages 187-197
  15. Eigenvalues of Endomorphisms
    Jörg Liesen, Volker Mehrmann
    Pages 199-212
  16. Polynomials and the Fundamental Theorem of Algebra
    Jörg Liesen, Volker Mehrmann
    Pages 213-226
  17. Cyclic Subspaces, Duality and the Jordan Canonical Form
    Jörg Liesen, Volker Mehrmann
    Pages 227-251
  18. Matrix Functions and Systems of Differential Equations
    Jörg Liesen, Volker Mehrmann
    Pages 253-269
  19. Special Classes of Endomorphisms
    Jörg Liesen, Volker Mehrmann
    Pages 271-293
  20. The Singular Value Decomposition
    Jörg Liesen, Volker Mehrmann
    Pages 295-302
  21. The Kronecker Product and Linear Matrix Equations
    Jörg Liesen, Volker Mehrmann
    Pages 303-310
  22. Back Matter
    Pages 311-324
    PDF

About this book

Introduction

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.

Keywords

Linear Algebra Matrices Echelon Form Gaussian Elimination Eigenvalues Linear Maps Vector Spaces Polynomials Fundamental Theorem of Algebra Jordan Canonical Form Matrix Functions Singular Value Decomposition Kronecker Product

Authors and affiliations

  1. 1.Institute of MathematicsTechnical University of BerlinBerlinGermany
  2. 2.Institute of MathematicsTechnical University of BerlinBerlinGermany

About the authors

Jörg Liesen's research interests are in Numerical Linear Algebra, Matrix Theory and Constructive Approximation, with a particular focus on the convergence and stability analysis of iterative methods. He is also interested in the history of Mathematics, and in particular of Linear Algebra. He is the recipient of several prizes and awards for his mathematical work, including the Householder Award, the Emmy Noether Fellowship and the Heisenberg Professorship of the DFG. He likes to teach and pursue Mathematics as a lively subject, connecting theory with an ever increasing variety of fascinating applications. 

Volker Mehrmann's research interests are in Numerical Mathematics, Control Theory, Matrix Theory as well as Scientific Computing. In recent years he has focused on the development and analysis of numerical methods for nonlinear eigenvalue problems and differential-algebraic systems with applications in many fields such as mechanical systems, electronic circuit simulation and acoustic field computations. He is co-editor-in-chief of the journal Linear Algebra and its Applications and editor of many other journals in Linear Algebra and Numerical Analysis. He believes that Mathematics has become a central prerequisite for the societal development of the 21st century and that mathematical methods play the key role in the modeling, simulation, control and optimization of all are as of technological development.

Bibliographic information

  • Book Title Linear Algebra
  • Authors Jörg Liesen
    Volker Mehrmann
  • Series Title Springer Undergraduate Mathematics Series
  • Series Abbreviated Title SUMS
  • DOI https://doi.org/10.1007/978-3-319-24346-7
  • Copyright Information Springer International Publishing Switzerland 2015
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-319-24344-3
  • eBook ISBN 978-3-319-24346-7
  • Series ISSN 1615-2085
  • Series E-ISSN 2197-4144
  • Edition Number 1
  • Number of Pages XI, 324
  • Number of Illustrations 22 b/w illustrations, 0 illustrations in colour
  • Topics Linear and Multilinear Algebras, Matrix Theory
  • Buy this book on publisher's site

Reviews

“This book begins with applied problems that are examined as the theory is developed. … Liesen and Mehrmann … present computations with matrix groups and rings, elementary matrices, echelon forms, rank, linear systems, determinants, and eigenvalues and eigenvectors before introducing vectors and vector spaces. … Each chapter ends with a set of exercises addressing both computation and theory. … Summing Up: Recommended. Upper-division undergraduates through professionals/practitioners.” (J. R. Burke, Choice, Vol. 53 (12), September, 2016)

“It provides a good introductory undergraduate course at an intermediate level … . From the beginning, the authors motivate the material with interesting examples … . the text includes short ‘MATLAB-Minutes’ which are exercises providing an informal introduction to the use of MATLAB in linear algebra, including hints about how care may be needed when working in finite precision.” (John D. Dixon, zbMATH 1334.15001, 2016)