1. Linear Equations 1.1 Introduction to Linear Systems 1.2 Matrices, Vectors, and Gauss-Jordan Elimination 1.3 On the Solutions of Linear Systems; Matrix Algebra   2. Linear Transformations 2.1 Introduction to Linear Transformations and Their Inverses 2.2 Linear Transformations in Geometry 2.3 Matrix Products 2.4 The Inverse of a Linear Transformation   3. Subspaces of Rn and Their Dimensions 3.1 Image and Kernel of a Linear Transformation 3.2 Subspace of Rn; Bases and Linear Independence 3.3 The Dimension of a Subspace of Rn 3.4 Coordinates   4. Linear Spaces 4.1 Introduction to Linear Spaces 4.2 Linear Transformations and Isomorphisms 4.3 The Matrix of a Linear Transformation   5. Orthogonality and Least Squares 5.1 Orthogonal Projections and Orthonormal Bases 5.2 Gram-Schmidt Process and QR Factorization 5.3 Orthogonal Transformations and Orthogonal Matrices 5.4 Least Squares and Data Fitting 5.5 Inner Product Spaces   6. Determinants 6.1 Introduction to Determinants 6.2 Properties of the Determinant 6.3 Geometrical Interpretations of the Determinant; Cramer's Rule   7. Eigenvalues and Eigenvectors 7.1 Diagonalization 7.2 Finding the Eigenvalues of a Matrix 7.3 Finding the Eigenvectors of a Matrix 7.4 More on Dynamical Systems 7.5 Complex Eigenvalues 7.6 Stability   8. Symmetric Matrices and Quadratic Forms 8.1 Symmetric Matrices 8.2 Quadratic Forms 8.3 Singular Values    Appendix A. Vectors Appendix B: Techniques of Proof Answers to Odd-numbered Exercises Subject Index Name Index

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.

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.