The articles in this collection discuss both the content of linear algebra courses and approaches to teaching such courses. The authors address elementary topics, such as row reduction, and more advanced topics, such as sparse matrices, iterative methods and pseudo-inverses. There are agreements:
- "Only under torture would I tell a student about Kramer's rule..."(Almon);
- "As an example of how familiarity with determinants can rot your brain...." (Axler).
There are, of course, also areas of disagreement, as when Dubinsky disagrees with the LACSG recommendations. There are well-known applications, such as Markov chains, and some not-so-well-known applications, such as Fisher's theorem on complete bipartite subgraphs (bicliques). There are articles from the users of matrix algebra: computer graphics, computer science and others. The articles together constitute a thoughtful, well-written, challenging and often entertaining discussion of this important area of mathematics.
PART I, "The Role of Linear Algebra," is a perfect way to begin this book. Alan Tucker gives a nice survey of the history of linear algebra with many historical notes on people such as Leibniz, (Wilhelm) Jordan, Babbage, von Neumann and Turing. He emphasizes the importance of linear algebra for its applicability and its role in computation. He discusses the pedagogical importance of linear algebra as a "very accessible geometrically based theory whose study serves as preparation to more abstract upper-division courses." (p.10). He also gives a nice overview of the history of the linear algebra course in the undergraduate curriculum.
In PART II, "Linear Algebra as Seen from Client Disciplines," four professionals who use linear algebra in their work discuss their views on linear algebra. We learn about applications in computer graphics, computer science, economics and engineering. "Matrix Algebra in Economics," written by Clopper Almon is quite enthusiastic about applications of linear algebra to statistics (least square models and related topics), to modeling an economy, to the maximization of functions of many variables subject to constraints and to dynamical systems defined by difference equations. Margaret Wright in "Linear Algebra for Computer Science Students" finds that most of her coworkers at AT&T Bell Laboratories use linear algebra and describes what she sees as the important topics for computer science majors. Rosemary Chang, in her article " A View from a Client Discipline, Computer Graphics," says that the more sophisticated the techniques in computer graphics, the more the need for a strong background in linear algebra. The concluding article in this section is by David P. Young, "Linear Algebra Use at Boeing: Implications for Undergraduate Education." He states that while there is not much use of linear algebra in the programming section at Boeing Computer services, there is in research and development. Useful topics include iterative methods, approximation theory, sparse matrix techniques, least squares, eigenvalues and eigenvectors and Gaussian elimination theory.
PART III, "The Teaching of Linear Algebra," is very useful. It shows that many of us who teach linear algebra face the same problems: we have all seen "the fog roll in" as we hit the more abstract portion of the course. This section includes a great discussion about the recommendations made by the Linear Algebra Curriculum Study Group (LACSG), including differing viewpoints by others. There is a discussion of various approaches used in the linear algebra classroom, and a useful article on conceptualization.
The first article, "Teaching Linear Algebra: Must the Fog Always Roll In?" by David Carlson, describes the common problems with ideas such as subspace, rank, basis, span and linear independence. He identifies the reasons he feels that the "fog rolls in" and makes some suggestions on how to deal with the "fog." The next article is "The Linear Algebra Curriculum Study Group Recommendations for the First Course in Linear Algebra." The recommendations are made with convincing justifications for their suggested core syllabus (included). Charles C. Cowen describes an interesting project that has come from problems faced in consulting work at Ford Motor Company in his article "A Project on Circles in Space." The project uses many basic concepts of linear algebra to determine if a set of points lie on a circle.
One of the most useful articles for those of us who are trying to include MATLAB in our courses is Jane M. Day's "Teaching Linear Algebra New Ways." This article describes the changes she has made in her linear algebra class and says it is a work still in progress. She has included MATLAB projects in her course, includes numerical issues, allows partners for computer work and has changed her style of teaching as well as testing. She makes suggestions of textbooks that include MATLAB exercises and has a lengthy bibliography, which I found most helpful.
In the well-written and well-documented article "Some Thoughts on a First Course in Linear Algebra at the College Level," Ed Dubinsky expresses his somewhat negative reaction to the recommendations of the LACSG and of David Carlson. He describes his philosophy on the way to approach undergraduate mathematics instruction in general and linear algebra in particular. Guershon Harel's article responds to the LACSG recommendations; he endorses the incorporation of technology and proposes the use of MATLAB. He has specific recommendations of his own including changing high school curriculum to enhance the learning of linear algebra on the college level, use of MATLAB in calculus in preparation for linear algebra, and how to teach proofs. Two articles follow dealing with specific topics in linear algebra, namely L-U factorization and iterative methods.
Robert Mena discusses the evolution of the undergraduate linear algebra course through the years in his article "Reflections (1988)". Gerald Porter in "Writing About Linear Algebra: Report on an Experiment" talks about a linear algebra course for non-mathematics majors in which he had the students write a ten page chapter to supplement the text material on subspaces, spanning sets, basis and dimension, lines, planes and hyperplanes. The final article in this section is "Scenes from Linear Algebra Classes" by Shlomo Vinner. He would like to see a change in teaching to an approach that emphasizes concepts, ideas and thought. He discusses and gives examples of conceptual and pseudoconceptual behavior of the students. He feels the place of proof in service courses is controversial but agrees with the LACSG recommendations emphasizing problem solving (including some proofs) and motivating applications. He concludes with the observation that there is a need to study cognitive issues as well as curricular issues.
Part IV, "Linear Algebra Exposition," contains very insightful and useful articles on the teaching of various topics in linear algebra. Sheldon Axler's "Down with Determinants!" takes the reader through a convincing and thought-provoking presentation of eigenvalues and eigenvectors without the use of determinants. He feels that the proper place to introduce determinants is late in the course (defining then as the products of eigenvalues), for use in the change of variables formula for multi-variable integrals. In "Subspaces and Echelon Forms" David Lay encourages linear algebra instructors to make sure that students can easily move between the explicit form of subspaces (all linear combinations of some vectors) and the implicit form (a solution set of a system of homogeneous linear equations). This paper describes how to do this using matrix echelon forms. Maron and Manwani in "A Geometric Interpretation of the Columns of the (Pseudo) Inverse of A" describes how the columns of the (pseudo)inverse of a matrix A can be used to project the i-th row of A on the span of the other rows. The final article in this section is "The Fundamental Theorem of Linear Algebra" by Gilbert Strang, which centers on figures illustrating the relations of the four important subspaces related to an mxn matrix A.
PART V, "Applications of Linear Algebra," completes the book by including six articles dealing with various applications. Many of the applications seem to require most of the material covered in a first linear algebra course and thus could only be presented near the end of the semester. This in no way lessens the value of these applications. In "Some Applications of Elementary Linear Algebra in Combinatorics" Brualdi and Quinn give three applications, two of which can be done without linear algebra (Fisher's inequality and Hall's Marriage Theorem), although using linear algebra techniques may simplify the solution. I especially liked the third application, Biclique Partitions, where linear algebra is essential to the solution. Clark and Datta present an enjoyable student-pleasing card trick that is based on invariance properties of certain matrix subspaces. Lange and Miller present an intriguing ladder game that is used in Japan to determine Christmas gifts. Gerald Porter in "Linear Algebra and Affine Planar Transformations" gives concrete ideas which are used in computer graphics and which are easily accessible to the linear algebra student. Again, this is a case of linear algebra simplifying the work even though it is not, strictly speaking, needed. This section ends with two wonderful articles by Gilbert Strang "Patterns in Linear Algebra" and "Graphs, Matrices, and Subspaces". In the first article, Strang gives two "remarkable families of matrices... which illustrate the central ideas of elimination and diagonalization and orthogonality... with numbers that make you smile." And, yes indeed, they make you smile. In his second article, Strang explains that "what y = f(x) is to calculus, matrices and subspaces are to linear algebra." He gives an application using connected directed graphs associated with incidence matrices. These matrices and their subspaces illustrate Kirkhoff's laws. It is a wonderful, accessible application.
Overall, this book is a source of much information and should be useful to many teachers of linear algebra. Even though many of the articles have been published elsewhere before, having them in one place and arranged by topic is an excellent idea. The client discipline articles give many examples of how linear algebra is used in the "real world," something students always want to know. The articles on the LACSG recommendations give us much food for thought, and the application section is filled with wonderful examples. The incorporation of technology is discussed in several articles and sources for projects are given. For those who teach linear algebra, this one's a "must get".
Rebecca Berg (firstname.lastname@example.org) is a professor of mathematics at Bowie State University.