Linear Algebra and Its Applications

Table of Contents

• Linear Equations in Linear Algebra
  • Introductory Example: Linear Models in Economics and Engineering
  • 1.1 Systems of Linear Equations
  • 1.2 Row Reduction and Echelon Forms
  • 1.3 Vector Equations
  • 1.4 The Matrix Equation Ax = b
  • 1.5 Solution Sets of Linear Systems
  • 1.6 Applications of Linear Systems
  • 1.7 Linear Independence
  • 1.8 Introduction to Linear Transformations
  • 1.9 The Matrix of a Linear Transformation
  • 1.10 Linear Models in Business, Science, and Engineering
  • Projects
  • Supplementary Exercises
• Matrix Algebra
  • Introductory Example: Computer Models in Aircraft Design
  • 2.1 Matrix Operations
  • 2.2 The Inverse of a Matrix
  • 2.3 Characterizations of Invertible Matrices
  • 2.4 Partitioned Matrices
  • 2.5 Matrix Factorizations
  • 2.6 The Leontief Input--Output Model
  • 2.7 Applications to Computer Graphics
  • 2.8 Subspaces of Rn
  • 2.9 Dimension and Rank
  • Projects
  • Supplementary Exercises
• Determinants
  • Introductory Example: Random Paths and Distortion
  • 3.1 Introduction to Determinants
  • 3.2 Properties of Determinants
  • 3.3 Cramer''s Rule, Volume, and Linear Transformations
  • Projects
  • Supplementary Exercises
• Vector Spaces
  • Introductory Example: Space Flight and Control Systems
  • 4.1 Vector Spaces and Subspaces
  • 4.2 Null Spaces, Column Spaces, and Linear Transformations
  • 4.3 Linearly Independent Sets; Bases
  • 4.4 Coordinate Systems
  • 4.5 The Dimension of a Vector Space
  • 4.6 Change of Basis
  • 4.7 Digital Signal Processing
  • 4.8 Applications to Difference Equations
  • Projects
  • Supplementary Exercises
• Eigenvalues and Eigenvectors
  • Introductory Example: Dynamical Systems and Spotted Owls
  • 5.1 Eigenvectors and Eigenvalues
  • 5.2 The Characteristic Equation
  • 5.3 Diagonalization
  • 5.4 Eigenvectors and Linear Transformations
  • 5.5 Complex Eigenvalues
  • 5.6 Discrete Dynamical Systems
  • 5.7 Applications to Differential Equations
  • 5.8 Iterative Estimates for Eigenvalues
  • 5.9 Markov Chains
  • Projects
  • Supplementary Exercises
• Orthogonality and Least Squares
  • Introductory Example: The North American Datum and GPS Navigation
  • 6.1 Inner Product, Length, and Orthogonality
  • 6.2 Orthogonal Sets
  • 6.3 Orthogonal Projections
  • 6.4 The Gram--Schmidt Process
  • 6.5 Least-Squares Problems
  • 6.6 Machine Learning and Linear Models
  • 6.7 Inner Product Spaces
  • 6.8 Applications of Inner Product Spaces
  • Projects
  • Supplementary Exercises
• Symmetric Matrices and Quadratic Forms
  • Introductory Example: Multichannel Image Processing
  • 7.1 Diagonalization of Symmetric Matrices
  • 7.2 Quadratic Forms
  • 7.3 Constrained Optimization
  • 7.4 The Singular Value Decomposition
  • 7.5 Applications to Image Processing and Statistics
  • Projects
  • Supplementary Exercises
• The Geometry of Vector Spaces
  • Introductory Example: The Platonic Solids
  • 8.1 Affine Combinations
  • 8.2 Affine Independence
  • 8.3 Convex Combinations
  • 8.4 Hyperplanes
  • 8.5 Polytopes
  • 8.6 Curves and Surfaces
  • Projects
  • Supplementary Exercises
• Optimization
  • Introductory Example: The Berlin Airlift
  • 9.1 Matrix Games
  • 9.2 Linear Programming-Geometric Method
  • 9.3 Linear Programming-Simplex Method
  • 9.4 Duality
  • Projects
  • Supplementary Exercises
• Finite-State Markov Chains (Online Only)
  • Introductory Example: Googling Markov Chains
  • 10.1 Introduction and Examples
  • 10.2 The Steady-State Vector and Google''s PageRank
  • 10.3 Communication Classes
  • 10.4 Classification of States and Periodicity
  • 10.5 The Fundamental Matrix
  • 10.6 Markov Chains and Baseball Statistics
• Appendices
• Uniqueness of the Reduced Echelon Form
• Complex Numbers