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Basic Linear Algebra

T. S. Blyth and E. F. Robertson
Springer Verlag
Publication Date: 
Number of Pages: 
[Reviewed by
Maria G. Fung
, on

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 ( 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.


akirak's picture

This book gives a thorough and rigorous treatment of linear algebra, which is what a first year student will expect to see on a linear algebra course from a British university. There are a number of numerical examples which lead nicely to the theory of linear algebra. The authors have hit the right balance between proofs of theorems and techniques to apply such theorems.

The ordering of the chapters is sensible, with the first four chapters covering matrices and linear equations before the more abstract work on vector spaces. The theory and manipulations on eigenvalues and eigenvectors is left towards the end of the book.

A great asset of the book is that it is portable and reasonably cheap at around £16 for students to buy and carry around in lectures and library. It is also good to see that brief solutions to most problems are at the back of the book. The only solutions omitted are the assignment problems which the lecturer can set as part of the coursework. There are sufficient exercises with good progression, and it is good to see a whole chapter devoted to a computer algebra package.

I have following reservations:

  • In the introduction to the book it is important to state why linear algebra is critical to the student's mathematical studies. It should say something like "after calculus the most useful mathematical tool ever developed is linear algebra because it brings the physical world within the scope of mathematics".
  • A book on linear algebra should have plenty of illustrations so that the student can envisage what is going on and these illustrations can be used to motivate him or her. This book has a severe lack of diagrams.
  • More words are required to motivate the student and soften the blow. Each chapter should have an introduction, a list of objectives and a summary. I follow the maxim "Tell them, at great length, what you are going to do. Do it, and then tell them what you have done'. The authors do not write in a way that will appeal to weaker students. It is far too succinct.
  • The word "basic' in the title is not appropriate for this book. A number of A level students cannot divide 10-7 by (1/2 x 104 ) even with a calculator. I can't see how students will cope with this book without serious input by a tutor.
  • The book is not interactive in any way. It seems to be a one way delivery from the authors to the student. A book like this should include some questions which will make the student think and arouse his/her anxiety. I could not find a single question in the text of the book for the reader. Clearly there are a number of problems for the student to tackle but I am referring to questions such as: 1. Why are matrices important? 2. How can we prove this theorem? 3. What approach are we going to use to solve this problem?
  • A more serious issue is that once the authors have covered a particular concept they expect the student to fully digest it. This is not my experience of students. I think a particular concept used in chapter 9 which was covered in chapter 2 say, needs to be signposted so that the student knows exactly where the idea was defined earlier in the book.
  • A less serious issue is that the authors use some very compact and complicated notation. It will difficult for first year students to follow some of this compact notation unless they have seen it before.

The authors use mathematical software, MAPLE 7, but it would have been better to integrate this into each chapter rather than bolt on a chapter at the end. Students will be more confident in using the software if it is used throughout the book.

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