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The Linear Algebra a Beginning Graduate Student Ought to Know

Jonathan S. Golan
Springer Verlag
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
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Sometimes the “why” of a book is at least as intriguing as the “what”. For me, this was especially true of The Linear Algebra a Beginning Graduate Student Ought to Know. The author has some pretty strong feelings on the subject. He says:

… in recent years the content of linear algebra courses required to complete an undergraduate degree in mathematics … has been depleted to the extent that it falls far short of what is needed for graduate study or for real-world application.

And he continues:

Students are not only less able to formulate or even follow mathematical proofs, they are also less able to understand the underlying mathematics of the numerical algorithms they must see. The resulting knowledge gap has led to frustration and recrimination on the part of both students and faculty alike, with each silently — and sometimes not so silently — blaming the other for the resulting state of affairs.

This book is intended to bridge that gap and presumably resolve the frustration and recriminations. (Who would have thought that linear algebra could inflame the passions like that?) I have virtually no recent experience with the preparation of incoming graduate students, but I am surprised that linear algebra would be a notable shortcoming. I might have thought that the focus of the first book like this to appear would have been analysis or abstract algebra, since many colleges have significantly reduced requirements in those areas. Perhaps I just haven’t seen those books yet.

The author has designed this book for several possible uses: for self study, as a text for advanced linear algebra for advanced undergraduate or first-year graduate students, as a reference book, or as study guide for Ph.D. qualifying exams.

The topics that the author considers are mostly standard, but his treatment of them is unusual. Probably most of us think that the core of linear algebra is about linear transformations of vector spaces over the real or complex fields. The author’s approach is more abstract. For example, he begins with vector spaces and algebras over an arbitrary field and works throughout — whenever possible — over an arbitrary field. After discussions of linear independence, dimension and linear transformations, he introduces the endomorphism algebra of a vector space.

When I read the opening sentences in the chapter on the algebra of square matrices (said to involve the algebraic structure of a set of matrices with entries in an associative unital F-algebra), I began to wonder seriously about the assumptions inherent in the book’s title. Of course, many examples and exercises in the text use actual real numbers. But the overall level of abstraction seems inappropriate for the intended audience, particularly if the purpose of the text is at least partly remedial.

Remarkably little attention is paid to numerical issues, especially given the author’s introductory comments about the importance of real-world applications and computational matrix theory. The basic matrix factorizations of great importance in numerical work get only passing attention. The author’s remark “Applied mathematicians … who design mathematical models often take considerable pains to avoid creating ill-conditioned systems” is simply naive and unhelpful. Ill-conditioned systems are unavoidable; the art is knowing how to deal with them.

The visual appearance of the text might well put off some readers. The pages are dense with text, and the text is full of exercises — more than a thousand of them. Indeed the pages are so full that things look all crammed together. When thumbnail photographs and capsule biographies of notable mathematicians are added at the bottom of some pages, it looks even worse.

There is such a mismatch here between the contents of this book and its intended audience that I could not recommend it. Although many consider it lowbrow, I have considerable respect for Schaum’s Outline of Linear Algebra. How much more than this does an incoming graduate student really need to know?

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

 1. Notation and terminology.- 2. Fields.- 3. Vector spaces over a field.- 4. Algebras over a field.- 5. Linear Dependence and Dimension.- 6. Linear Transformations.- 7. The endomorphism algebra of a vector space.- 8. Representation of linear transformations by matrices.- 9. The algebra of square matrices.- 10. Systems of linear equations.- 11.Determinants.- 12. Eigenvalues and eigenvectors.- 13. Krylov subspaces.- 14. The dual space.- 15. Inner product spaces.- 16. Orthogonality.- 17. Selfadjoint endomorphisms.- 18. Unitary and normal endomorphisms.- 19. Moore-Penrose pseudoinverses.- 20. Bilinear transformations and forms.