billcruise/linearalgebra

Linear Algebra examples in R

LinearAlgebra

Linear Algebra examples in R Selected solutions to example problems from Linear Algebra and its Applications, 5th ed. by David C. Lay, Steven R. Lay, and Judi McDonald

Glossary of Terms

Chapter 1: Linear Equations in Linear Algebra

1. Systems of Linear Equations
2. Row Reduction and Echelon Forms
3. Vector Equations
4. The Matrix Equation Ax = b
5. Solution Sets of Linear Systems
6. Applications of Linear Systems
7. Linear Independence
8. Introduction to Linear Transformations
9. The Matrix of a Linear Transformation
10. Linear Models in Business, Science, and Engineering

Chapter 2: Matrix Algebra

1. Matrix Operations
2. The Inverse of a Matrix
3. Characterizations of Invertible Matrices
4. Partitioned Matrices
5. Matrix Factorizations
6. The Leontief Input-Output Model
7. Applications to Computer Graphics
8. Subspaces of Rⁿ
9. Dimension and Rank

Chapter 3: Determinants

1. Introduction to Determinants
2. Properties of Determinants
3. Cramer's Rule, Volume, and Linear Transformations

Chapter 4: Vector Spaces

1. Vector Spaces and Subspaces
2. Null Spaces, Columns Spaces, and Linear Transformations
3. Linearly Independent Sets; Bases
4. Coordinate Systems
5. The Dimension of a Vector Space
6. Rank
7. Change of Basis
8. Applications to Difference Equations
9. Application to Markov Chains

Chapter 5: Eigenvalues and Eigenvectors

1. Eigenvectors and Eigenvalues
2. The Characteristic Equation
3. Diagonalization
4. Eigenvectors and Linear Transformations
5. Complex Eigenvalues
6. Discrete Dynamical Systems
7. Applications to Differential Equations
8. Iterative Estimates for Eigenvalues

Chapter 6: Orthogonality and Least Squares

1. Inner Product, Length, and Orthogonality
2. Orthogonal Sets
3. Orthogonal Projections
4. The Gram-Schmidt Process
5. Least-Squares Problems
6. Applications to Linear Models
7. Inner Product Spaces
8. Applications of Inner Product Spaces

Chapter 7: Symmetric Matrices and Quadratic Forms

1. Diagonalization of Symmetric Matrices
2. Quadratic Forms
3. Constrained Optimization
4. The Singular Value Decomposition
5. Applications to Image Processing and Statistics

Chapter 8: The Geometry of Vector Spaces

1. Affine Combinations
2. Affine Independence
3. Convex Combinations
4. Hyperplanes
5. Polytopes
6. Curves and Surfaces

Chapter 9: Optimization

1. Matrix Games
2. Linear Programming - Geometric Method
3. Linear Programming - Simplex Method
4. Duality

Chapter 10: Finite-State Markov Chains

1. Introduction and Examples
2. The Steady-State Vector and Google's PageRank
3. Communication Classes
4. Classification of States and Periodicity
5. The Fundamental Matrix
6. Markov Chains and Baseball Statistics