Before Starting Linear Algebra
The outline of this curriculum is based on the David Lay Linear Algebra Textbook
How to Read the Textbook(s)
Chapter 1: Linear Equations in Linear Algebra
- Systems of Linear Equations
- Row Reduction and Echelon Forms
- Vector Equations
- The Matrix Equation Ax = b
- Solution Sets of Linear Systems
- Applications of Linear Systems
- Linear Independence
- Introduction to Linear Transformations
- The Matrix of a Linear Transformation
- Linear Models in Business, Science, and Engineering
Chapter 2: Matrix Algebra
- Introductory Example: Computer Models in Aircraft Design
- Matrix Operations
- The Inverse of a Matrix
- Characterizations of Invertible Matrices
- Partitioned Matrices
- Matrix Factorizations
- The Leontief Input–Output Model
- Applications to Computer Graphics
- Subspaces of R^n
- Dimension and Rank
- Supplementary Exercises
Chapter 3: Determinants
- Introductory Example: Random Paths and Distortion
- Introduction to Determinants
- Properties of Determinants
- Cramer’s Rule, Volume, and Linear Transformations
- Supplementary Exercises
Chapter 4: Vector Spaces
- Introductory Example: Space Flight and Control Systems
- Vector Spaces and Subspaces
- Null Spaces, Column Spaces, and Linear Transformations
- Linearly Independent Sets; Bases
- Coordinate Systems
- The Dimension of a Vector Space
- Rank
- Change of Basis
- Applications to Difference Equations
- Applications to Markov Chains
- Supplementary Exercises
Chapter 5: Eigenvalues and Eigenvectors
- Introductory Example: Dynamical Systems and Spotted Owls
- Eigenvectors and Eigenvalues
- The Characteristic Equation
- Diagonalization
- Eigenvectors and Linear Transformations
- Complex Eigenvalues
- Discrete Dynamical Systems
- Applications to Differential Equations
- Iterative Estimates for Eigenvalues
- Supplementary Exercises
Chapter 6: Orthogonality and Least Squares
- Introductory Example: The North American Datum and GPS Navigation
- Inner Product, Length, and Orthogonality
- Orthogonal Sets
- Orthogonal Projections
- The Gram–Schmidt Process
- Least-Squares Problems
- Applications to Linear Models
- Inner Product Spaces
- Applications of Inner Product Spaces
- Supplementary Exercises
Chapter 7: Symmetric Matrices and Quadratic Forms
- Introductory Example: Multichannel Image Processing
- Diagonalization of Symmetric Matrices
- Quadratic Forms
- Constrained Optimization
- The Singular Value Decomposition
- Applications to Image Processing and Statistics
- Supplementary Exercises
Chapter 8: The Geometry of Vector Spaces
- Introductory Example: The Platonic Solids
- Affine Combinations
- Affine Independence
- Convex Combinations
- Hyperplanes
- Polytopes
- Curves and Surfaces
Chapter 9: Optimization (Online)
- Introductory Example: The Berlin Airlift
- Matrix Games
- Linear Programming—Geometric Method
- Linear Programming—Simplex Method
- Duality
Chapter 10: Finite-State Markov Chains (Online)
- Introductory Example: Googling Markov Chains
- Introduction and Examples
- The Steady-State Vector and Google’s PageRank
- Communication Classes
- Classification of States and Periodicity
- The Fundamental Matrix
- Markov Chains and Baseball Statistics