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Linear Algebra: Concepts and Methods

Martin Anthony and Michele Harvey
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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At the risk of opening a nasty can of worms, I want to go on record with a complaint. My university’s mathematics department, like so many others across the country (and beyond), is infected by the virus of consumerism: it’s all about warm bodies in the classroom and, hand in glove therewith, making erstwhile majors’ courses available to fellow travelers of various stripes. Thus, my department’s complex analysis course is no longer predicated on a transitional course on how to do proofs, and allows kids in with only third-semester calculus under their belts. This tends to cause, for instance, an eruption of engineering majors trying to squeeze in a Mathematics minor.

Along similar lines, but perhaps not quite as egregiously, there is the case of linear algebra, where we encounter a rather similar mix of students, but generally with even less mathematical maturity. What this translates to is a course in which examples often take the place of proofs, the theorems having been stated in an accessible form, and calculation routines play a dominant role. Of course we all need to know these routines, but not all that long ago they were part of the appropriate lower-level course, with linear algebra proper coming later and being largely devoted to theorems and proofs. In my own pre-historic case, I recall the upper division linear algebra course at UCLA, taught by the late Chitikila Musili, visiting from India: Professor Musili proved loads of theorems very elegantly and when indicated referred us to Halmos’s Finite Dimensional Vector Spaces: no holding back, no excuses, all real mathematics.

That was then, this is now. What do we do to address the current crop of students, of mathematics majors even, so many of whom are actually averse to proving theorems? (I’ve heard senior majors say that they do not like “theory courses”! Isn’t that along the same lines as an English major saying “yeah, dude, like, it’s the reading of those big books that I just can’t wrap my head around, man”?) Well, we make do, and we do the best we can. In this semester’s linear algebra course I’m not doing as well as I would like, having chosen a book that is in fact far too “theoretical” for my audience — a lesson painfully learned. Something different is needed, and I think that the book under review may be a solution to this problem.

For what it’s worth, Anthony and Harvey’s Linear Algebra: Concepts and Methods is something of an exemplar of a certain Realpolitik approach to the current pedagogical situation, but, happily, the book also covers the subject more thoroughly than its competitors do. Much more so than the other books I’ve seen lately, this book succeeds in presenting the subject properly to fellow travelers (e.g. economists!) to the extent this is reasonable, and doesn’t skimp on actual mathematics.

A large part of the reason for this is that the authors teach at the London School of Economics and aim their text at a broad group of students, including undergraduates in various degree programs as well as “students studying at a distance,” in other words, a broader spectrum than is usually the case. Accordingly, the authors stress that their text “sets out to introduce and explain linear algebra” even as they populate the text very heavily with worked examples and problems largely aimed at mathematical beginners. Thus, the book has theorems and proofs (not like Halmos’s book qua relative austerity, but good mathematical proofs nonetheless, very suitable for rookies willing to work), as well as a wealth of illustrations and discussions leading the reader to progressively improved perceptions about the subject and increased facility with the requisite manipulations and computations.

The book is big, in fact bigger than most linear algebra books at this level, weighing in at over 500 pages. But its pace is appropriately slow and much of the book is taken up by such things as sections on “learning outcomes” and “comments on activities” commensurate with the existence of a lot of “activities” in the respective preceding chapters. (Each of the book’s thirteen chapters closes with this sort of thing: not my cup of tea, as the cousins across the pond might say, but it’s all but ubiquitous, isn’t it?) The fare is orthodox and, as I already suggested, thorough, and at times goes one step beyond — consider, for example, the book’s twelfth chapter on direct sums and projections: Anthony and Harvey include discussions on orthogonal complements and projections as well as “minimising the distance to a subspace” and the method of least squares.

Yes, I think that Linear Algebra: Concepts and Methods is bound to be a very successful book in today’s market. I for one intend to use it the next time I’m at bat in the linear algebra line-up.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Preliminaries: before we begin
1. Matrices and vectors
2. Systems of linear equations
3. Matrix inversion and determinants
4. Rank, range and linear equations
5. Vector spaces
6. Linear independence, bases and dimension
7. Linear transformations and change of basis
8. Diagonalisation
9. Applications of diagonalisation
10. Inner products and orthogonality
11. Orthogonal diagonalisation and its applications
12. Direct sums and projections
13. Complex matrices and vector spaces
14. Comments on exercises