Added values

These seven undergraduate texts all have some geometric flavour. I shall treat them in three categories according to their goals and intended readership.

Introduction to Linear Algebra and Linear Algebra: Ideas and Applications are well-written texts suitable for a level one unit in linear algebra. These texts strike a balance between theory and applications in the sense that the theoretical framework is interwoven carefully with the development of tools and algorithms, which are then used in realistic applications. The student is introduced to abstract mathematics and also gains an appreciation of the utility of linear algebra. Although the subject coverage in these books is standard, the treatment is up to date and makes full use of technology.

The first book, which reaches a slightly higher level than the second, features clear highlighted definitions, historical snippets, and an appendix on the use of Matlab. Also included in this fourth edition are Matlab projects. In the second book, the useful chapter summaries deserve a mention, as does the glossary, which defines over a hundred concepts. There are also some very well thought-out exercises including a selection of freestanding Matlab problems.

The two algebra texts, Introduction to Algebra and An Introductory Course in Commutative Algebra , complement each other well. The first forms an ideal basis for first and second units in abstract algebra, for second or third-level courses in group theory and ring theory, and for supporting courses in Galois theory and coding theory. The exposition is clear, the prerequisites are minimal, and it covers a lot of ground. The book develops the central themes well and applies them to coding theory and to Galois theory, the latter being used to provide an "almost-algebraic" proof of the fundamental theorem of algebra. Particularly useful is chapter seven, which further develops the theory of groups and rings and gives an illuminating introduction to the unifying frameworks of universal algebra and of category theory, enabling the student to take a wider contextual view of the subject and pointing the way to advanced study. There are many examples and plenty of exercises as well as an interesting further reading section and a website containing more problems, further topics, a list of misprints, and links to other algebra sites.

An Introductory Course in Commutative Algebra describes the theory of commutative rings and some applications to number theory, geometry and finite field theory. The book could be used as the course text for an undergraduate unit at second, third or fourth level, the prerequisites being some basic group theory - say up to the first isomorphism theorem - and a first course in linear algebra. It would also be a useful supplement to a course in general algebra or number theory. Although the text is rather concise, the pace is not too fast and some very important theorems are proved. The authors work through plenty of concrete computations and provide 130 exercises (with solutions for about half). The book is carefully organised with good motivational introductions to each chapter and helpful summary sections.

The final three books are a move away from single-subject textbooks. They highlight the unity of mathematics, in particular the subtle interplay between algebra, arithmetic and geometry, and will be relevant to a wide range of readers. Indeed, Geometry: Plane and Fancy and Numbers and Geometry could, after careful reading, be understood by someone armed with only school algebra and analytic geometry. On the other hand, sections of Glimpses of Algebra and Geometry require an undergraduate background for full comprehension.

Geometry focuses on Euclidean, hyperbolic and elliptic geometry, with complex numbers and quaternions being introduced to describe isometries. The book might be used as a course book for a non-Euclidean geometry unit, but would serve better as background reading or as a general interest text.Although this is the least technical of the seven books, many results are supplied with proofs, and there are non-trivial problems distributed throughout.

In the excellent Numbers and Geometry , we find a clear account of a diverse collection of subjects - number theory, geometry, algebra and analysis - all linked together using the themes of linear and quadratic equations. It could be used by undergraduates and teachers wishing to experience a unified treatment of many deep and classic ideas, as a text for conventional units in number theory or geometry at level two or higher,or for an overview unit. There are many sets of exercises, each of which is accompanied by a commentary. The discussion sections at the ends of chapters tie together major themes in a mathematical and historical context.

Climbing even further into the realms of high-quality, insightful exposition, we find Glimpses of Algebra and Geometry , which would make essential reading for the third or fourth year of a MMath programme where it is desirable to promote a global viewpoint of mathematics. Indeed, the author facilitates the transition to postgraduate studies in topology and geometry. The book includes 170 superb problems (with solution hints for 43 of them) and recommends useful websites. I know of no other text at this level that develops such an impressive range of ideas with such clarity, and I thoroughly recommend it.

Tim Swift is lecturer in mathematics,University of Southampton New College