Preface 11emPreliminaries 1.11emVectors and Matrices 1.21emMATLAB 1.31emSpecial Kinds of Matrices 1.41emThe Geometry of Vector Operations 21emSolving Linear Equations 2.11emSystems of Linear Equations and Matrices 2.21emThe Geometry of Low-Dimensional Solutions 2.31emGaussian Elimination 2.41emReduction to Echelon Form 2.51emLinear Equations with Special Coefficients 2.61emUniqueness of Reduced Echelon Form 31emMatrices and Linearity 3.11emMatrix Multiplication of Vectors 3.21emMatrix Mappings 3.31emLinearity 3.41emThe Principle of Superposition 3.51emComposition and Multiplication of Matrices 3.61emProperties of Matrix Multiplication 3.71emSolving Linear Systems and Inverses 3.81emDeterminants of 22 Matrices 41emSolving Linear Differential Equations 4.11emA Single Differential Equation 4.21em*Rate Problems 4.31emUncoupled Linear Systems of Two Equations 4.41emCoupled Linear Systems 4.51emThe Initial Value Problem and Eigenvectors 4.61emEigenvalues of 22 Matrices 4.71emInitial Value Problems Revisited 4.81em*Markov Chains 51emVector Spaces 5.11emVector Spaces and Subspaces 5.21emConstruction of Subspaces 5.31emSpanning Sets and MATLAB 5.41emLinear Dependence and Linear Independence 5.51emDimension and Bases 5.61emThe Proof of the Main Theorem 61emClosed Form Solutions for Planar ODEs 6.11emThe Initial Value Problem 6.21emClosed Form Solutions by the Direct Method 6.31emSimilar Matrices and Jordan Normal Form 6.41emSinks, Saddles, and Sources 6.51em*Matrix Exponentials 6.61em*The Cayley Hamilton Theorem 6.71em*Second Order Equations 71emDeterminants and Eigenvalues 7.11emDeterminants 7.21emEigenvalues 7.31emReal Diagonalizable Matrices 7.41em*Existence of Determinants 81emLinear Maps and Changes of Coordinates 8.11emLinear Mappings and Bases 8.21emRow Rank Equals Column Rank 8.31emVectors and Matrices in Coordinates 8.41em*Matrices of Linear Maps on a Vector Space 91emLeast Squares 9.11emLeast Squares Approximations 9.21emLeast Squares Fitting of Data 101emOrthogonality 10.11emOrthonormal Bases and Orthogonal Matrices 10.21emGram-Schmidt Orthonormalization Process 10.31emThe Spectral Theory of Symmetric Matrices 10.41em*QR Decompositions 111em*Matrix Normal Forms 11.11emSimple Complex Eigenvalues 11.21emMultiplicity and Generalized Eigenvectors 11.31emThe Jordan Normal Form Theorem 11.41em*Markov Matrix Theory 11.51em*Proof of Jordan Normal Form 121emMatlab Commands Index
