Features

New Features in the Fourth Edition

• Diagonalization and eigenvalues are now introduced early (in Chapter 3 using only determinants and matrix inverses), and so can be presented to beginning students in their first course. This unique feature, suggested originally by the University of Calgary Engineering Faculty, opens up a number of important applications. For example, ecological models are an excellent motivation for the students because they can relate them to the extinction of species and see the relevance to the real world.

• Chapter 5 is a new "bridging chapter" in which concepts such as subspaces, spanning, independence, dimension and linear transformations, are introduced into Rⁿ. This avoids "hitting the wall" for students who first see these ideas in the setting of an abstract vector space, and serves as a good stopping point for a first half-course at the second year level. (It provides an early, rigorous treatment of rank and diagonalization.)

• The last section of Chapter 5 introduces linear transformations in Rⁿ, building on examples in R³ such as reflections and rotations. It requires only Chapters 2 and 4 (the material in section 5.1 can be easily supplied directly), and provides a much needed geometrical view of matrix algebra.

• Increased flexibility for the instructor to choose different routes through the material (see the Section Dependency Diagram following the Chapter Summaries).

• Numerous computational exercises with no book answer have been revised, allowing them to be used again for assignments.

• The material on block multiplication (in section 2.2) has been shortened to emphasize only those features that occur later in the book. This eliminates tedious examples that are time-consuming in class.

• The proof (in section 2.3) of the fundamental characterizations of invertibility has been simplified and shortened, and does not require elementary matrices. This simplifies teaching of beginning students.

• Section 2.5 on elementary matrices has been shortened by about 30 percent, which makes it easier to cover when needed later in the course.

• New applications of diagonalization to linear recurrences (in section 3.5) and population growth (in section 3.6) are now included. These are unique to this book because they occur early in the text (requiring only matrix inverses and determinants) and give examples of the use of linear methods that the students can understand.

• An application to chemical reactions has been included at the end of Chapter 1, adding to applications to network flows and electrical networks in Chapter 1 and economic models and Markov chains in Chapter 2. These give real-world examples early in the book which motivate students in other disciplines, many of whom do not yet really understand why they are studying linear algebra.

• The section on positive definite matrices and the Cholesky factorization has been completely rewritten; it is now both shorter and easier to follow. The section on QR-factorization has also been rewritten.

• The appendix on linear programming has been moved to the text specific website at www.mcgrawhill.ca/college/nicholson.

Other Features

• Presentation of techniques in examples, with emphasis on concrete computations and on the algorithmic nature of some techniques, allows students to master new skills readily. The text has more than 330 solved examples that cover the basic techniques, illustrate the central idea, and are keyed to the exercises in the book.

• A wide variety of exercises, which start with routine computational problems and progress to more theoretical exercises, help students develop skills in an appropriate, logically paced fashion.

• End-of-Chapter applications showing how linear algebra clarifies and solves problems provide relevance for students. These are placed at the end of the chapter containing the necessary technologies.

• Answers to even-numbered computational exercises and selected others enable students to check the accuracy of their computation immediately.