Blyth and Robertson's Basic Linear Algebra sets as its goal "to provide in a reasonably concise and readable form a rigorous first course that covers all of the material on linear algebra to which every student of mathematics should be exposed at an early stage." This book covers the foundational topics of matrices, systems of linear equations, vector spaces, linear mappings, determinants, eigenvalues and eigenvectors, and the minimal polynomial. It also includes a brief introduction to the Linear Algebra package of MAPLE 7. More advanced topics, such as Fourier expansion, canonical forms, and the structure of normal matrices over the real and complex field, are reserved for the sequel, Further Linear Algebra, by the same authors. Readers of Basic Linear Algebra are expected to be familiar with calculus and to have a solid experience reading and writing proofs.
The authors perform what they set out to do with this book. Basic Linear Algebra is elegantly concise. It is written in a clean example-theorem-proof style, where the story is told by judicious choice and ordering of the examples and concepts, from the concrete and familiar to the general and abstract. For instance, Chapter 6 dedicated to linear maps, starts off with some concrete examples that are used to develop the notions of Image and Kernel spaces and to prove the Dimension Theorem. These concepts then are used to show that any two finite dimensional vector spaces of the same dimension are isomorphic and even to exhibit an example of a linear map on an infinite dimensional vector space (one built from real sequences that are zero in all but finitely many spaces) that is surjective but not injective.
The exercises in the book form a rich resource for linear algebra instruction for undergraduate students. There are exercises that are interspersed throughout the text and serve to ensure students' understanding, and there are more exercises at the end of each chapter. In turn, the end of the chapter set consists of standard exercises, supplementary exercises and assignment exercises. All the exercises are wonderfully selected in terms of topics and mathematical exposure to important ideas from different fields of mathematics. For example, one could learn about nilpotent matrices and then get a glimpse at the exponential and logarithmic mappings for these matrices. In another exercise, the reader finds out what an exact sequence is and proves the vanishing of the alternating sums of the dimensions of the vector spaces in such a sequence.
This book is clearly geared towards students of mathematics (rather, perhaps, than to users of linear algebra from other disciplines). The manner of exposition is terse yet very clear and elegant, and it is apparent the authors are algebraists. Concepts tend to be introduced axiomatically; the authors usually begin by asserting a theorem that there exists some (perhaps unique) object satisfying certain axioms, and then constructing such an object as part of the proof. This style is very precise and rigorous, but may challenge some readers who are not yet accustomed to this mode of discourse. For instance, Theorem 1.2 on page 4 asserts that "There is a unique m x n matrix M such that, for every m x n matrix A, A + M = A". Then, in the proof, they construct an m x n matrix whose entries are all 0, and then they prove that it is the unique matrix with this property. After that, they define the resulting matrix to be the "m x n zero matrix". As another (quite inspiring) example, Chapter 8, which is dedicated to determinants, starts off with an axiomatic definition of a determinantal mapping as a multilinear, alternating, and 1-preserving mapping. Then the authors inductively construct a determinantal mapping on n x n matrices from a determinantal mapping on (n-1) x (n-1) matrices (disposing of the 2 x 2 case separately) and then establish uniqueness by setting up basic concepts from the theory of permutation groups.
In the end, I suspect that this book's somewhat abstract and spare expository style may stretch the mathematical sophistication level of many American undergraduates who are taking a first course in linear algebra, but I would relish the opportunity to offer an honors course in linear algebra from it or to suggest this book for supplementary readings and projects for motivated students. I believe Basic Linear Algebra is a valuable reference for an inspirationally elegant and streamlined algebraic development of the foundational ideas of the subject.
Maria G. Fung (firstname.lastname@example.org) is assistant professor of mathematics at Western Oregon University. Her interests are the mathematical preparation of K-8 teachers and the representation theory of Lie groups.