Linear Algebra with Applications

Features Linear transformations are introduced early in the text to make the discussion of matrix operations more meaningful and easier to navigate. Visualization and geometrical interpretation are emphasized extensively throughout the text. An abundance of problems, exercises, and applications help students assess their understanding and master the material. Abstract concepts are introduced gradually throughout the text. Major ideas are carefully developed at various levels of generality before the student is introduced to abstract vector spaces. Discrete and continuous dynamical systems are used as a motivation for eigenvectors and as a unifying theme thereafter. Fifty to sixty True/False questions conclude every chapter, testing conceptual understanding and encouraging students to read the text. Historical problems from ancient Chinese, Indian, Arabic, and early European sources add diversity to the selection of exercises. Rotations, reflections, projections, and shears are used throughout to illustrate new ideas. Multiple perspectives on the kernel of a matrix are presented, with a stronger emphasis on the relations among the columns. Commutative diagrams enhance students' conceptual understanding of the matrix of a linear transformation. These diagrams enable students to visualize the relations between linear transformations. A more conceptual approach to the QR factorization is presented, in terms of a change-of-basis. New To This Edition A large number of exercises have been added to the problem sets, from the elementary to the challenging and from the abstract to the applied. For example, there are quite a few new exercises on “Fibonacci Matrices” and their eigenvectors and eigenvalues. Throughout the text, the author added an ongoing discussion of the mathematical principles behind search engines—and the notion of PageRank in particular—with dozens of examples and exercises. Besides being an interesting and important contemporary application of linear algebra, this topic allows for an early and meaningful introduction to dynamical systems, one of the main themes of this text, naturally leading up to a discussion of diagonalization and eigenvectors. A new appendix offers a brief discussion of the proof techniques of Induction and Contraposition. Hundreds of improvements, such as offering hint in a challenge problem, for example, or choosing a more sensible notation in a definition. Offering the most geometric presentation available, Linear Algebra with Applications, Fifth Edition emphasizes linear transformations as a unifying theme. This elegant textbook combines a user-friendly presentation with straightforward, lucid language to clarify and organize the techniques and applications of linear algebra. Exercises and examples make up the heart of the text, with abstract exposition kept to a minimum. Exercise sets are broad and varied and reflect the author’s creativity and passion for this course. This revision reflects careful review and appropriate edits throughout, while preserving the order of topics of the previous edition.