1. Linear Equations in Linear AlgebraIntroductory Example: Linear Models in Economics and Engineering1.1 Systems of Linear Equations1.2 Row Reduction and Echelon Forms1.3 Vector Equations1.4 The Matrix Equation Ax = b1.5 Solution Sets of Linear Systems1.6 Applications of Linear Systems1.7 Linear Independence1.8 Introduction to Linear Transformations1.9 The Matrix of a Linear Transformation1.10 Linear Models in Business, Science, and EngineeringSupplementary Exercises 2. Matrix AlgebraIntroductory Example: Computer Models in Aircraft Design2.1 Matrix Operations2.2 The Inverse of a Matrix2.3 Characterizations of Invertible Matrices2.4 Partitioned Matrices2.5 Matrix Factorizations2.6 The Leontief Input-Output Model2.7 Applications to Computer Graphics2.8 Subspaces of Rn2.9 Dimension and RankSupplementary Exercises 3. DeterminantsIntroductory Example: Random Paths and Distortion3.1 Introduction to Determinants3.2 Properties of Determinants3.3 Cramer's Rule, Volume, and Linear TransformationsSupplementary Exercises 4. Vector SpacesIntroductory Example: Space Flight and Control Systems4.1 Vector Spaces and Subspaces4.2 Null Spaces, Column Spaces, and Linear Transformations4.3 Linearly Independent Sets; Bases4.4 Coordinate Systems4.5 The Dimension of a Vector Space4.6 Rank4.7 Change of Basis4.8 Applications to Difference Equations4.9 Applications to Markov ChainsSupplementary Exercises 5. Eigenvalues and EigenvectorsIntroductory Example: Dynamical Systems and Spotted Owls5.1 Eigenvectors and Eigenvalues5.2 The Characteristic Equation5.3 Diagonalization5.4 Eigenvectors and Linear Transformations5.5 Complex Eigenvalues5.6 Discrete Dynamical Systems5.7 Applications to Differential Equations5.8 Iterative Estimates for EigenvaluesSupplementary Exercises 6. Orthogonality and Least SquaresIntroductory Example: Readjusting the North American Datum6.1 Inner Product, Length, and Orthogonality6.2 Orthogonal Sets6.3 Orthogonal Projections6.4 The Gram-Schmidt Process6.5 Least-Squares Problems6.6 Applications to Linear Models6.7 Inner Product Spaces6.8 Applications of Inner Product SpacesSupplementary Exercises 7. Symmetric Matrices and Quadratic FormsIntroductory Example: Multichannel Image Processing7.1 Diagonalization of Symmetric Matrices7.2 Quadratic Forms7.3 Constrained Optimization7.4 The Singular Value Decomposition7.5 Applications to Image Processing and StatisticsSupplementary Exercises 8. The Geometry of Vector SpacesIntroductory Example: The Platonic Solids8.1 Affine Combinations8.2 Affine Independence8.3 Convex Combinations8.4 Hyperplanes8.5 Polytopes8.6 Curves and Surfaces 9. Optimization (Online Only)Introductory Example: The Berlin Airlift9.1 Matrix Games9.2 Linear Programming-Geometric Method9.3 Linear Programming-Simplex Method9.4 Duality 10. Finite-State Markov Chains (Online Only)Introductory Example: Google and Markov Chains10.1 Introduction and Examples10.2 The Steady-State Vector and Google's PageRank10.3 Finite-State Markov Chains10.4 Classification of States and Periodicity10.5 The Fundamental Matrix10.6 Markov Chains and Baseball Statistics AppendicesA. Uniqueness of the Reduced Echelon FormB. Complex Numbers