Linear Algebra with Applications – Gareth Williams – 8th Edition

Part 1 Linear Equations, Vectors, and Matrices
1 Linear Equations and Vectors
1.1 Matrices and Systems of Linear Equations
1.2 Gauss-Jordan Elimination
1.3 The Vector Space Rn
1.4 Subspaces of Rn
1.5 Basis and Dimension in Rn
1.6 Dot Product, Norm, Angle, and Distance
1.7 Curve Fitting, Electrical Networks, and Traffic Flow
Chapter 1 Review Exercises

2 Matrices and Linear Transformations
2.1 Addition, Scalar Multiplication, and Multiplication of Matrices
2.2 Properties of Matrix Operations
2.3 Symmetric Matrices and Seriation in Archaeology
2.4 The Inverse of a Matrix and Cryptography
2.5 Matrix Transformations, Rotations, and Dilations
2.6 Linear Transformations, Graphics, and Fractals
2.7 The Leontief Input-Output Model in Economics
2.8 Markov Chains, Population Movements, and Genetics
2.9 A Communication Model and Group Relationships in Sociology
Chapter 2 Review Exercises

3 Determinants and Eigenvectors
3.1 Introduction to Determinants
3.2 Properties of Determinants
3.3 Determinants, Matrix Inverses, and Systems of Linear Equations
3.4 Eigenvalues and Eigenvectors
3.5 Google, Demography, Weather Prediction, and Leslie Matrix Models
Chapter 3 Review Exercises

Part 2 Vector Spaces
4 General Vector Spaces
4.1 General Vector Spaces and Subspaces
4.2 Linear Combinations of Vectors
4.3 Linear Independence of Vectors
4.4 Properties of Bases
4.5 Rank
4.6 Projections, Gram-Schmidt Process, and QR Factorization
4.7 Orthogonal Complement
4.8 Kernel, Range, and the Rank/Nullity Theorem
4.9 One-to-One Transformations and Inverse Transformations
4.10 Transformations and Systems of Linear Equations
Chapter 4 Review Exercises

5 Coordinate Representations
5.1 Coordinate Vectors
5.2 Matrix Representations of Linear Transformations
5.3 Diagonalization of Matrices
5.4 Quadratic Forms, Difference Equations, and Normal Modes
Chapter 5 Review Exercises

6 Inner Product Spaces
6.1 Inner Product Spaces
6.2 Non-Euclidean Geometry and Special Relativity
6.3 Approximation of Functions and Coding Theory
6.4 Least Squares Solutions
Chapter 6 Review Exercises

Part 3 Numerical Linear Algebra
7 Numerical Methods
7.1 Gaussian Elimination
7.2 The Method of LU Decomposition
7.3 Practical Difficulties in Solving Systems of Equations
7.4 Iterative Methods for Solving Systems of Linear Equations
7.5 Eigenvalues by Iteration and Connectivity of Networks
7.6 The Singular Value Decomposition
Chapter 7 Review Exercises

8 Linear Programming
8.1 A Geometrical Introduction to Linear Programming
8.2 The Simplex Method
8.3 Geometrical Explanation of the Simplex Method
Chapter 8 Review Exercises

Appendices
A Cross Product
B Equations of Planes and Lines in Three-Space
C Graphing Calculator Manual
C1 Reduced Echelon Form of a Matrix
C2 Matrix Operations
C3 Powers of a Matrix
C4 Transpose of a Matrix
C5 Inverse of a Matrix
C6 Determinant of a Matrix
C7 Summary of Formats for Row Operations
D MATLAB Manual
D1 Entering and Displaying a Matrix (Section 1.1)
D2 Solving Systems of Linear Equations (Sections 1.1, 1.2, 1.7)
D3 Dot Product, Norm, Angle, Distance (Section 1.6)
D4 Matrix Operations (Sections 2.1–2.3)
D5 Computational Considerations (Section 2.2)
D6 Inverse of a Matrix (Section 2.4)
D7 Solving Systems of Equations Using Matrix Inverse (Section 2.4)
D8 Cryptography (Section 2.4)
D9 Transformations Defined by Matrices (Sections 2.5, 2.6)
D10 Fractals (Section 2.6)
D11 Leontief I/O Model (Section 2.7)
D12 Markov Chains (Sections 2.8, 3.5)
D13 Digraphs (Section 2.9)
D14 Determinants (Sections 3.1–3.3)
D15 Cramer’s Rule (Section 3.3)
D16 Eigenvalues and Eigenvectors (Sections 3.4, 3.5)
D17 Linear Combinations, Dependence, Basis, Rank (Sections 1.3, 4.2–4.5)
D18 Projection, Gram-Schmidt Orthogonalization (Section 4.6)
D19 QR Factorization (Section 4.6)
D20 Kernel and Range (Section 4.8)
D21 Inner Product, Non-Euclidean Geometry (Sections 6.1, 6.2)
D22 Space-Time Travel (Section 6.2)
D23 Pseudoinverse and Least Squares Curves (Section 6.4)
D24 LU Decomposition (Section 7.2)
D25 Condition Number of a Matrix (Section 7.3)
D26 Jacobi and Gauss-Seidel Iterative Methods (Section 7.4)
D27 Singular Value Decomposition (Section 7.6)
D28 The Simplex Method in Linear Programming (Section 8.2)
D29 Cross Product (Appendix A)
D30 MATLAB Commands, Functions, and M-Files
D31 The Linear Algebra with Applications Toolbox M-Files