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
4.21emRate 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
4.81emMarkov 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
6.51emMatrix Exponentials
6.61emThe Cayley Hamilton Theorem
6.71emSecond Order Equations
71emDeterminants and Eigenvalues
7.11emDeterminants
7.21emEigenvalues
7.31emReal Diagonalizable Matrices
7.41emExistence 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
8.41emMatrices 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
10.41emQR Decompositions
111emMatrix Normal Forms
11.11emSimple Complex Eigenvalues
11.21emMultiplicity and Generalized Eigenvectors
11.31emThe Jordan Normal Form Theorem
11.41emMarkov Matrix Theory
11.51em*Proof of Jordan Normal Form
121emMatlab Commands
Index