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Linear algebra. A textbook about the theory with a view toward practice. (Lineare Algebra. Ein Lehrbuch über die Theorie mit Blick auf die Praxis.) (German)
Bachelorkurs Mathematik. Wiesbaden: Vieweg+Teubner (ISBN 978-3-8348-0081-7/pbk). x, 302~p. (2011).
This book is based on lectures given by the authors at the TU Chemnitz and the TU Berlin and covers the linear algebra required for the Bachelor’s degree in mathematical subjects at German universities. The authors have chosen the matrix concept as the central object of their approach to linear algebra, as it is ubiquitous in situations where linear algebra is applied. The first third of the book (100 pages) covers basics and all the important properties of matrices, after which abstract vector spaces and linear transformations are studied. The chapter headings are (in translation): 1. Everyday linear algebra, 2. Mathematical basics, 3. Algebraic structures, 4. Matrices, 5. Echelon normal form and rank, 6. Systems of linear equations, 7. Determinants, 8. Characteristic polynomial and eigenvalues, 9. Vector spaces, 10. Linear transformations, 11. Linear and bilinear forms, 12. Euclidean and unitary spaces, 13. Adjoint linear transformations, 14. Eigenvalues of endomorphisms, 15. Polynomials and the fundamental theorem of algebra, 16. Cyclic subspaces, duality and the Jordan normal form, 17. Matrix functions and systems of differential equations, 18. Special classes of endomorphisms, 19. The singular value decomposition, 20. The Kronecker product and linear matrix equations. The first chapter discusses several real-life applications of linear algebra that are developed further in later chapters. The authors have also included so-called MATLAB-Minutes with which readers can recapitulate certain results with their PC. Each chapter (except the first) concludes with a set of predominantly theoretical problems. The book is carefully written and will be a pleasure to read for any student used to reading mathematical texts. In the reviewer’s opinion it is not suitable for the majority of economics students.
Reviewer: Rabe von Randow (Bonn)
Classification: H15 H65