LECTURE NOTES OF WILLIAM CHEN
# LINEAR ALGEBRA

## SECTION A --- ELEMENTARY TOPICS

### Chapter 1 : LINEAR EQUATIONS >>

### Chapter 2 : MATRICES >>

### Chapter 3 : DETERMINANTS >>

### Chapter 4 : VECTORS >>

## SECTION B --- INTRODUCTION TO SOME ALGEBRAIC STRUCTURES

### Chapter 5 : INTRODUCTION TO VECTOR SPACES >>

### Chapter 6 : VECTOR SPACES ASSOCIATED WITH MATRICES >>

### Chapter 7 : EIGENVALUES AND EIGENVECTORS >>

## SECTION C --- FURTHER TOPICS

### Chapter 8 : LINEAR TRANSFORMATIONS >>

### Chapter 9 : REAL INNER PRODUCT SPACES >>

### Chapter 10 : ORTHOGONAL MATRICES >>

### Chapter 11 : APPLICATIONS OF REAL INNER PRODUCT SPACES >>

### Chapter 12 : COMPLEX VECTOR SPACES >>

This set of notes has been compiled over a period of more than 30 years. Some chapters were used in various forms and on many occasions between 1981 and 1990 by the author at Imperial College, University of London. The remaining chapters were written in Sydney.

The material has been organized in such a way to create a single volume suitable to take the reader to a reasonable level of linear algebra. Chapters 1 - 4 cover very basic material. The concept of vector spaces is then introduced in Chapters 5 - 7. More advanced topics, including the concept of linear transformations from one vector space to another and the concept of inner products, are covered in Chapters 8 - 12.

To read the notes, click the links below for connection to the appropriate PDF files.

The material is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, the documents may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners.

- Introduction
- Elementary Row Operations
- Row Echelon Form
- Reduced Row Echelon Form
- Solving a System of Linear Equations
- Homogeneous Systems
- Application to Network Flow
- Application to Electrical Networks
- Application to Economics
- Application to Chemistry
- Application to Mechanics

- Introduction
- Systems of Linear Equations
- Inversion of Matrices
- Application to Matrix Multiplication
- Finding Inverses by Elementary Row Operations
- Criteria for Invertibility
- Consequences of Invertibility
- Application to Economics
- Matrix Transformation on the Plane
- Application to Computer Graphics
- Complexity of a Non-Homogeneous System
- Matrix Factorization
- Application to Games of Strategy

- Introduction
- Determinants for Squares Matrices of Higher Order
- Some Simple Observations
- Elementary Row Operations
- Further Properties of Determinants
- Application to Curves and Surfaces
- Some Useful Formulas
- Further Discussion

- Introduction
- Vectors in 2-Space
- Vectors in 3-Space
- Vector Products
- Scalar Triple Products
- Application to Geometry in 3-Space
- Application to Mechanics

- Real Vector Spaces
- Subspaces
- Linear Combination
- Linear Independence
- Basis and Dimension

- Introduction
- Row Spaces
- Column Spaces
- Rank of a Matrix
- Nullspaces
- Solution of Non-Homogeneous Systems

- Introduction
- The Diagonalization Problem
- Some Remarks
- An Application to Genetics

- Euclidean Linear Transformations
- Linear Operators on the Plane
- Elementary Properties of Euclidean Linear Transformations
- General Linear Transformations
- Change of Basis
- Kernel and Range
- Inverse Linear Transformations
- Matrices of General Linear Transformations
- Change of Basis
- Eigenvalues and Eigenvectors

- Euclidean Inner Products
- Real Inner Products
- Angles and Orthogonality
- Orthogonal and Orthonormal Bases
- Orthogonal Projections

- Introduction
- Eigenvalues and Eigenvectors
- Orthonormal Diagonalization

- Least Squares Approximation
- Quadratic Forms
- Real Fourier Series

- Complex Inner Products
- Unitary Matrices
- Unitary Diagonalization