- David C. Lay, Steven R. Lay, and Judi J. McDonald, Linear Algebra and Its Applications, 5th Edition (Pearson, 2016).
- Gilbert Strang, Linear Algebra and Its Applications, 4th Edition (Cengage, 2006).
- W. Keith Nicholson, Linear Algebra with Applications (Open Edition), Version 2019, Revision A (Lyryx Learning, 2019).
Part I — Foundations of Linear Systems
- Introduction to Linear Models
- Linear equations in economics, physics, and engineering (Lay)
- The geometry of linear equations (Strang)
- Systems of equations and network flow (Nicholson)
- Row Reduction and Gaussian Elimination
- Echelon and reduced echelon forms (Lay & Nicholson)
- Gauss–Jordan elimination and matrix notation (Strang)
- Elementary operations and consistency
- Matrix Representations and Vector Equations
- The matrix equation ( A\mathbf{x} = \mathbf{b} )
- Vector equations and span (Lay)
- Matrix–vector multiplication as transformation (Nicholson)
- Solution Sets and Linear Independence
- Homogeneous and non-homogeneous systems
- Parametric vector form (Lay)
- Nullspace interpretation and column space (Strang)
Part II — Matrix Algebra and Applications
- Matrix Operations and Algebraic Properties
- Addition, scalar multiplication, transposition (Nicholson)
- Associativity and distributive laws
- Block and partitioned matrices (Lay)
- Matrix Inverses and Factorizations
- Inverse and elementary matrices (Lay & Nicholson)
- LU, QR, and other decompositions (Strang)
- Applications to computer graphics and input–output models
- Subspaces, Rank, and Dimension
- Column space, nullspace, row space, and rank (Lay & Strang)
- Basis and dimension theorems
- The Fundamental Theorem of Linear Algebra (Strang)
Part III — Determinants and Eigenstructure
- Determinants and Their Properties
- Cofactor expansion and properties (Nicholson & Strang)
- Cramer’s Rule and geometry of volume (Lay)
- Applications in linear transformations
- Eigenvalues and Eigenvectors
- Characteristic equations and diagonalization (Lay & Strang)
- Defective matrices and similarity (Nicholson)
- Applications to differential and difference equations
- Complex Eigenvalues and Dynamical Systems
- Discrete dynamical systems (Lay)
- Continuous systems and ( e^{At} ) (Strang)
- Markov chains and recurrences (Nicholson)
Part IV — Vector Spaces and Transformations
- Vector Spaces and Subspaces
- Abstract vector spaces and ( \mathbb{R}^n ) (Lay & Nicholson)
- Spanning, basis, and coordinate systems
- Dimension theorem and change of basis
- Linear Transformations
- Mapping, kernel, and image (Lay & Nicholson)
- Matrix representation and composition (Strang)
- Isomorphisms and applications to differential equations
- Geometry of Vector Spaces
- Affine and convex combinations (Lay)
- Graphs and networks (Strang)
- Projections and planes in ( \mathbb{R}^3 ) (Nicholson)
Part V — Orthogonality, Optimization, and Applications
- Inner Product Spaces and Orthogonality
- Inner product, norms, and orthogonal projections (Lay & Strang)
- Gram–Schmidt and orthogonal bases
- Applications to least squares and regression models
- Symmetric and Positive Definite Matrices
- Spectral theorem and orthogonal diagonalization (Nicholson & Strang)
- Quadratic forms and constrained optimization (Lay)
- Principal Component Analysis (Nicholson)
- Singular Value Decomposition (SVD)
- Fundamental subspaces and pseudoinverse (Nicholson)
- Image compression and numerical stability
- Connections to rank and orthogonality
Part VI — Advanced Topics and Modern Methods
- Numerical Linear Algebra
- Condition numbers and matrix norms (Strang)
- Iterative methods for ( A\mathbf{x} = \mathbf{b} )
- Computational aspects of eigenvalue algorithms
- Optimization and Game Theory
- Linear programming and the simplex method (Strang)
- Duality and matrix games
- Network flow and resource allocation
- Markov Chains and Stochastic Models
- Steady-state vectors and Google’s PageRank (Lay)
- Transition matrices and communication classes (Nicholson)
- Canonical and Jordan Forms
- Jordan decomposition (Nicholson & Strang)
- Block triangular and invariant subspaces
- Applications in control theory
Educational Use Disclaimer
This unified Table of Contents has been created for educational and comparative study purposes only under the principles of fair use for academic instruction and synthesis of open and copyrighted materials.
- David C. Lay et al., Linear Algebra and Its Applications, ©2016 Pearson Education.
- Gilbert Strang, Linear Algebra and Its Applications, ©2006 Wellesley–Cambridge Press and Cengage.
- W. Keith Nicholson, Linear Algebra with Applications (Open Edition), ©2019 Lyryx Learning, under a Creative Commons BY-NC-SA license.
All rights to the original content remain with their respective copyright holders.
This document is intended solely for instructional alignment, research, and cross-text integration within mathematics education.
