Publisher: John Wiley (2008)Details: 480 pages, Hardcover
Edition: 3
Price: $110.00
ISBN: 9780470178843
Category: Monograph
Topics: Linear Algebra
Richard C. Penney
Publisher: John Wiley (2008)
Details: 480 pages, Hardcover
Edition: 3
Price: $110.00
ISBN: 9780470178843
Category: Monograph
Topics: Linear Algebra
[Reviewed by P. N. Ruane, on 09/15/2008]
To say that a particular book is ‘readable’ or ‘student-friendly’ is, of course, a matter of subjective comparison. For example, for the very first treatise on linear algebra, by Hermann Grassmann (1844), comparison was obviously impossible. But such was the inscrutable style of his exposition that many of the greatest mathematicians of his generation found it incomprehensible.
A century later, however, Paul Halmos’ book Finite Dimensional Vector Spaces (1947) became the very first university textbook devoted solely to linear algebra. And yet, although easygoing compared to Grassmann’s innovative book, most postgraduate students would find it very hard going as an introductory text (for instance, chapter 10 deals with calculus on linear manifolds, and there is hardly an exercise in the book).
Linear algebra was a rare component of undergraduate mathematics until the 1960s, and the publication of Finite Mathematical Structures, by Kemeny, Snell, Thompson and Merkel, (1959) may have precipitated its emergence as one of the most commonly occurring components in undergraduate mathematics. Appropriately, in 1965 the CUPM recommendation suggested that linear algebra be a course in the undergraduate curriculum and gave guidelines for course content.
Fifty years later, a quick search on the Barnes and Noble web site will come up with over 700 books with the phrase ‘linear algebra’ included in the title (perhaps with duplications, such as the listing of both hard and soft back versions of a particular book). Consequently, the process of choosing a book on this subject may be a more complex process than reading that by Paul Halmos.
It also seems to me that books on linear algebra are getting thicker and thicker as years go by — which is partly due, no doubt, to the wider range of themes they now cover. For instance, coverage of matrices has long been integrated into books like this one by Richard Penney, and in some depth at that. Also, many books cover a wide range of applications; in this book these include graph theory, circuit theory, computing, traffic flow, predator-prey problems, harmonic oscillators, geometry, numerical techniques, economics, and many more.
In the preface, the author says that this book not just about techniques, theorems and proofs, but about ideas. What this means is that the central concepts of linear algebra are not suddenly unloaded upon the reader, but they emerge gradually due to the careful structuring of the text. Another way of saying this is that the material is developed heuristically by means of examples and exercises that constitute about one third of the book’s 480 pages. Not only that, but Richard Penney writes very clearly, and there are no overly extended sections of prose in the whole book.
As for the exercises, they vary greatly in their nature and difficulty. Many sets are preceded by a True/False collection. Others consist of problems that are graded from routine to longer and more open-ended questions. In addition, there are about twenty computer projects dispersed across the books eight chapters, which are designed so that students undertake them on an independent basis using MATLAB (or Maple).
Regarding the contents, each of the chapters concludes with a summary of its main ideas, and, with the exception of duality, they cover nearly every topic that one would meet in an undergraduate introductory course on linear algebra. In fact, the amount of material here is possibly sufficient for a two-semester course. Moreover, due to the book’s readability and the organisation of the material, it is also highly suited to self-study.
Being a third edition, this book may be familiar to many readers of MAA reviews, so here’s a summary of the main changes:
Finally, I think it was Bertrand Russell who said something like ‘the bigger the book the greater its malign influence’, but that doesn’t apply to this one by Richard Penney — although I do I feel that it more readily applies to book reviews, so I’ll conclude by saying that, if Barnes and Noble have better books on linear algebra than this one, then the process of choosing from their vast collection is far more complex than I thought.
Peter Ruane was introduced to linear algebra by means of the book by Kemeny, Snell et al in 1964. However, he also referred to the relevant chapters of Birkhoff and MacLane, whose book A Survey of Modern Algebra (1940) contained the first undergraduate treatment of linear algebra.