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**Linear Algebra II. Applications and concepts for students in mathematics and physics.
(Lineare Algebra II. Anwendungen und Konzepte für Studierende der Mathematik und Physik.)**
*(German)*
Zbl 1373.15003

Heidelberg: Springer Spektrum (ISBN 978-3-662-53347-5/pbk; 978-3-662-53348-2/ebook). xv, 248 p. (2017).

This is the second of a two-volume text on linear algebra. The first one comprises a more or less standard introduction for students of mathematics and physics and was reviewed in [S. Waldmann, Lineare Algebra I. Die Grundlagen für Studierende der Mathematik und Physik. Heidelberg: Springer Spektrum (2017; Zbl 1357.15001)]; not quite standard because the author starts with basic set theory and then introduces semigroups, monoids, groups, rings and fields and uses these concepts throughout the text. This flavor is maintained in the second volume. Here, however, the topics to be included are not generally canonical – there is far more choice. In keeping with the fact that the author is specifically addressing the students mentioned above, he has chosen to include the following four topics, covered in four chapters. The first is an important application of linear algebra: the solution of linear differential equations with constant coefficients. To this end the matrix exponential function is treated in depth. Two examples are studied: the harmonic oscillator and coupled harmonic oscillators. The next topic is an abstract one, namely the formation of quotients in algebra, here the quotients of groups, rings and vector spaces. The treatment is modern, using commutative diagrams and short exact sequences, and closes with a discussion of dual vector spaces. The third topic forms, in the author’s own words, the heart of the book: a thorough introduction to the tensor product via its universal property. It is by far the longest chapter, starting with multilinear mappings, then defining the tensor product of vector spaces and of linear mappings and proving their many properties, including the role of dual spaces. Thankfully the author has included a section on Bourbaki’s “debauchery of indices” which many physicists still use for doing calculations. This part ends with a section on symmetric and antisymmetric tensors and one on tensor algebras. The fourth and last part deals with bilinear forms and quadrics. The first hurdle is Sylvester’s law of inertia for real symmetric bilinear forms. The next section deals with antisymmetric bilinear forms and the theorem of Darboux. The final section deals with real quadrics, including the principal axis theorem in Euclidean spaces. For \(n=2\) the conic sections are mentioned, but not the quadric surfaces for \(n=3\), which is a pity as they are so closely related to the conic sections and are easily depicted. There are more than 100 exercises, most of them theoretical and introducing further facts. In summary, the topics which the author has chosen in this second volume are important ones and readers looking for a thorough treatment of one or other of them will gain a great deal from this book.

Reviewer: Rabe von Randow (Bonn)

### MSC:

15-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra |

15A69 | Multilinear algebra, tensor calculus |

15A63 | Quadratic and bilinear forms, inner products |

34A26 | Geometric methods in ordinary differential equations |

34A30 | Linear ordinary differential equations and systems |

15A16 | Matrix exponential and similar functions of matrices |