Part I: INTRODUCTORY LINEAR ALGEBRA 1 Linear Equations and Matrices      3 1.1 Linear Systems      3 1.2 Matrices      11 1.3 Dot Product and Matrix Multiplication      18 1.4 Properties of Matrix Operations      35 1.5 Solutions of Linear Systems of Equations      47 1.6 The Inverse of a Matrix      69 2 Determinants      91 2.1 Definition and Properties      91 2.2 Cofactor Expansion and Applications      103 2.3 Determinants from a Computational Point of View      118 3 Vectors in R 2 and R n      123 3.1 Vectors in the Plane      123 3.2 n - Vectors      141 3.3 Introduction to Linear Transformations      158 3.4 Computer Graphics (Optional)      171 3.5 Cross Product in R 3 (Optional)      179 3.6 Lines and Planes      186 4 Real Vector Spaces      197 4.1 Real Vector Spaces      197 4.2 Subspaces      203 4.3 Linear Independence      213 4.4 Basis and Dimension      224 4.5 Homogeneous Systems      237 4.6 The Rank of a Matrix and Applications      244 4.7 Coordinates and Change of Basis      255 4.8 Orthonormal Bases in R n      269 4.9 Orthogonal Complements      277 5 Eigenvalues and Eigenvectors      291 5.1 Diagonalization      291 5.2 Diagonalization of Symmetric Matrices      312 6 Linear Transformations and Matrices      327 6.1 Definition and Examples      237 6.2 The Kernel and Range of a Linear Transformation      334 6.3 The Matrix of a Linear Transformation      345 Cumulative Review of Part I      366 Part II: APPLICATIONS 7 Linear Programming      371 7.1 The Linear Programming Problem: Geometric Solution      371 7.2 The Simplex Method      389 7.3 Duality      408 8 Applications      417 8.1 Graph Theory      417 8.2 Electrical Circuits      434 8.3 Markov Chains      439 8.4 Least Squares      450 8.5 Linear Economic Models      461 8.6 Differential Equations (Calculus Required)      469 8.7 The Fibonacci Sequence      480 8.8 Quadratic Forms      484 8.9 Conic Sections      493 8.10 Quadric Surfaces      501 8.11 The Theory of Games      511 Part III: NUMERICAL LINEAR ALGEBRA 9 Numerical Linear Algebra      535 9.1 Error Analysis      535 9.2 Linear Systems      538 9.3 LU-Factorization (Optional)      549 9.4 QR-Factorization (Optional)      556 9.5 Eigenvalues and Eigenvectos      561 Part IV: MATLAB FOR LINEAR ALGEBRA 10 MATLAB for Linear Algebra      573 10.1 Input and Output in MATLAB      573 10.2 Matrix Operations in MATLAB      579 10.3 Matrix Powers and Some Special Matrices      582 10.4 Elementary Row Operations in MATLAB      585 10.5 Matrix Inverses in MATLAB      595 10.6 Vectors in MATLAB      596 10.7 Applications of Linear Combinations in MATLAB      598 10.8 Linear Transformations in MATLAB 602 10.9 MATLAB Command Summary      604 Appendix A Complex Numbers      A1 A.1 Complex Numbers      A1 A.2 Complex Numbers in Linear Algebra      A10 Appendix B Further Directions      A23 B.1 Inner Product Spaces (Calculus Required)      A23 B.2 Composite and Invertible Linear Transformations      A32 Answers to Odd-Numbered Exercises and Chapter Tests      A41 Index      I1