Rezensionen
Rezensionen
Jürgen Hausen
Lineare Algebra II
Buchbesprechung
This booklet is the second volume of the author´s lecture notes on linear algebra, which grew out of a course he taught at the University of Tübingen, Germany. The first volume was published in [(2007; Zbl 1126.15001)] and covered the basic material of the first semester in linear algebra at German universities, including vector spaces, linear maps, matrices and determinants, diagonalization, Euclidean spaces, unitary vector spaces, and some relevant basic abstract algebra. The present second volume contains the (more advanced) material of the second semester in linear algebra for undergraduates, emphasizing the following standard topics:
1. Some more algebra (groups, commutative rings, ideals and factor rings);
2. Division in rings (divisibility in integral domains, Euclidean rings, factorial rings and prime decomposition);
3. Modules (modules and module homomorphisms, free modules, free modules over principal ideal domains, torsion modules, and length of a module);
4. Modules over Euclidean rings (matrices over Euclidean rings, structure of finitely generated modules over Euclidean rings, elementary divisors and primary decomposition);
5. Normal forms of endomorphisms (annihilator of a module, the minimal polynomial of a vector space endomorphism, cyclic vector spaces, elementary divisors of an endomorphism, the Cayley-Hamilton theorem, the rational normal form of matrices over a field, the Jordan normal form and the Jordan decomposition, effective computation of normal forms);
6. Bilinear and multilinear algebra (bilinear forms over vector spaces, symmetric bilinear forms, multilinear maps of vector spaces, tensor products and their universal property, bases of tensor spaces, alternating multilinear maps of vector spaces, exterior products and their universal property, bases of exterior products).
Each of the 21 sections of the book comes with its own series of related exercises, ranging from instructive concrete examples and applications to abstract-theoretical supplements of the main text. Generally, the material is divided into six parts, each of which is subdivided into several sections. Each section corresponds to a lecture of 90 minutes, thus faithfully reflecting the course lectures for the convenience of the students. Also, the author has successfully managed to combine a very clear and rigorous presentation of the standard material with a maximum of pleasant conciseness, clarifying remarks, gentle reminders, and illustrating examples, thereby making the current classroom notes a fairly self-contained and user-friendly source for effective study. Especially the included effective algorithms and computational aspects with regard to normal forms of endomorphisms and matrices are certainly of particular value for undergraduate students.
Werner Kleinert (Berlin)
