Hannah Robbins has written a successful Functional Linear Algebra text. The book contains an in-depth discussion of linear functions and guides students to the concepts of vectors and matrix operations. The order in which Robbins presents many topics is non-traditional. Elementary row operations are discussed early on (in Section 2.3), and vector spaces appear at the very beginning. Vector spaces, subspaces, independence, and span are all discussed over the real numbers. Applications to population modeling in biology, modeling molecular structures in chemistry, statistics, coordinate systems in machining, and economics modeling are also discussed. For example, Chapter 0 starts with a balancing chemical reaction example and progresses through recording connections using 0’s or 1’s.

I will admit that when I first saw that linear combinations and span were covered so early in the text (in Section 1.2), I was concerned about how effectively they would be presented. However, Robbins presents many simple yet effective examples of these topics, first in \( R^{2} \) and later in \( R^{3} \). Example 3 on pages 27-28, which concerns the linear combinations of three restaurant lunch specials, was a particularly intriguing application. Also, the pictures of the spanning examples provide a useful visual perspective on the concept of span.

Chapter 2 presents matrices along with linear functions. In Example 5 on pages 62-64, Robbins returns to a problem from Chapter 0 on the five stages of the life cycle of the smooth coneflower. This provides a solid foundational example in which students identify the coefficients of a function within a matrix; in this case \( f: R^{5} \rightarrow R^{5} \). Chapter 3 discusses vector spaces and change of bases, not only for \( R^{n} \) but also for spaces of polynomials, complex numbers, and matrices. One example I enjoyed was Example 7 on page 236:

Find the kernel of the map \( f: C \rightarrow M_{22} \) given by:

\( f(a+bi) = \left[ \begin{array}{cc} a-2b & 0 \\ 0 & -2a+4b \\ \end{array} \right] \)

Another very interesting application is presented on page 246:

Suppose a population has demographic matrix \( A \) so that if \( x \) is the population vector for one year, then \( Ax \) is the population vector for the next year. If \( x \) is an eigenvector of \( A \) with eigenvalue \( \lambda = 0.75 \), does this population grow or die out in the long run?

This problem involves taking powers of \( A \) and using the fact that \( \lim_{k \rightarrow \infty} (0.75)^{k}=0 \), and is a nice application of eigenvalues to population growth.

After reading this text I’m strongly considering using it the next time I teach a course on matrices. Even though the order of topics is a little different from the one I have traditionally used in class, I feel as though this text would work and would benefit my students. If you are looking for a new linear algebra text for a class of students that have had a solid foundation of calculus, I would highly consider this text.

Peter Olszewski, M.S., is a Mathematics Lecturer at The Pennsylvania State University, The Behrend College, an editor for Larson Texts, Inc. in Erie, PA, and is the 362nd Chapter Advisor of the Pennsylvania Alpha Beta Chapter of Pi Mu Epsilon. His Research fields are in mathematics education, Cayley Color Graphs, Markov Chains, and mathematical textbooks. He can be reached at

pto2@psu.edu or

www.personal.psu.edu/pto2. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar and bass, reading, gardening, traveling, and painting landscapes.