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Exercises in Linear Algebra

Luis Barreira and Claudis Valls
World Scientific
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Tushar Das
, on

Barreira and Valls’s book of problems would fall somewhere between Springer’s Problem Books and Schaum’s Outlines, perhaps closer to the latter. There are six chapters that comprise an even balance between 200 solved problems and 200 proposed problems (without solutions). The chapters cover Matrices and Vectors, Determinants, Vector Spaces, Linear Transformations, Inner Products and Norms, and Eigenvalues and Eigenvectors.

Unlike the book series mentioned above, the chapters jump straight in to solved exercises without any review of the definitions or concepts involved. This drawback makes the book difficult to read for a beginning student. For instance, #3.29 introduces the notation \(S \oplus R\) without explicitly mentioning the name of the concept involved, and expects the reader to know the definition. The lack of an index or a glossary of terms adds to this defect. The authors could remedy such a situation by reminding the reader of the definition of a direct sum within the solution of the exercise, and then go on to prove that the conditions one needs to check are satisfied.

The format of placing the solutions right after each problem makes it difficult for a student to try any of the solved problems on their own without reading their solution. An idea (I have seen used in the Springer Problem Book series) is to have the problems collected in one section, have a second section with brief hints, and then a third section with full solutions.

Though the authors claim that “Through a systematic discussion of 200 exercises, important concepts and topics are reviewed”, the progression of problems did not seem very systematic and appeared uneven in a number of places. For instance, here is a progression of about seven consecutive problems from the chapter on Vector Spaces: Find whether the set formed by all polynomials having only terms of even degree is a vector space. Find (various) equations of straight lines and planes given some points and direction vectors. Prove that the set of all symmetric \(n \times n\) matrices is a subspace of the vector space of all \(n \times n\) matrices.

There are a handful of problems that did come as a surprise. For example, here is #2.25: Given a matrix \(A \in M_{3 \times 3}(\mathbb{R})\), verify that the function \(p(\lambda) = \det (A - \lambda I)\) satisfies \[ p'(0)= \frac{\mathrm{tr}(A^2) - (\mathrm{tr} A)^2}{2}. \] I was stumped when I first looked at this formula, and it took me some time before seeing a context in which to place the problem (Maclaurin-Taylor series and Newton-Girard formulae). This led naturally to a strategy for generalizing beyond the \(3 \times 3\) case and including higher order terms. The book provides little guidance to the student regarding how to view this problem. It appears “out of nowhere”, followed by a tedious solution (take an arbitrary \(3 \times 3\) matrix and compute both sides of the identity). The result then reappears over 100 pages later in a problem (#6.19) asking the student to show that \[ \displaystyle \frac{\mathrm{tr}(A^2) - (\mathrm{tr} A)^2}{2} = \lambda_1\lambda_2 + \lambda_1\lambda_3+ \lambda_2\lambda_3, \] where the \(\lambda_i\) are eigenvalues of \(A \in M_{3 \times 3}(\mathbb{R})\) counted with multiplicities. The connection(s) between these problems could have been made more explicit by the inclusion of some narrative glue or better-informed cross-referencing. The algebraically inclined reader of this discussion may be reminded of (or induced to reconsider) problem #9.3 in Serre’s Linear Representations of Finite Groups.

The book gives the impression of being a collection of all the exercises culled from a standard undergraduate first course in linear algebra whose guiding text or notes are absent. The undergraduate student who wants “to see more solved examples” could easily form part of the intended target audience. Instructors may find the text to be a useful compendium for mining “fresh” questions for assessment. Though I was not impressed overall, the book may have the potential to become a useful problem book for students in a future revised edition.

Tushar Das is an Assistant Professor of Mathematics at the University of Wisconsin–La Crosse.

  • Matrices and Vectors
  • Determinants
  • Vector Spaces
  • Linear Transformations
  • Inner Products and Norms
  • Eigenvalues and Eigenvectors