Linear Algebra
by WWL Chen
This set of notes has been compiled
over a period of more than 25 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. All 12 chapters have been in
use at Macquarie University since 1997.
The material has been organized in
such a way to create a single volume suitable for use in the algebra
half of the units MATH135, MATH136, MATH132, MATH133 and MATH235
at Macquarie University. The following is the suggested order
for the presentation of the material:
MATH135 and MATH132:
MATH136 and MATH133:
MATH235:
- Chapters 8, 9, 10, 11 and 12.
To read the notes, click the chapters
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.
Chapter 1: LINEAR EQUATIONS
- 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
Chapter 2: MATRICES
- 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
Chapter 3: DETERMINANTS
- 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
Chapter 4: VECTORS
- Introduction
- Vectors in 2-Space
- Vectors in 3-Space
- Vector Products
- Scalar Triple Products
- Application to Geometry in 3-Space
- Application to Mechanics
Chapter 5: INTRODUCTION
TO VECTOR SPACES
- Real Vector Spaces
- Subspaces
- Linear Combination
- Linear Independence
- Basis and Dimension
Chapter 6: VECTOR SPACES
ASSOCIATED WITH MATRICES
- Introduction
- Row Spaces
- Column Spaces
- Rank of a Matrix
- Nullspaces
- Solution of Non-Homogeneous Systems
Chapter 7: EIGENVALUES
AND EIGENVECTORS
- Introduction
- The Diagonalization Problem
- Some Remarks
- An Application to Genetics
Chapter 8: LINEAR TRANSFORMATIONS
- 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
Chapter 9: REAL INNER
PRODUCT SPACES
- Euclidean Inner Products
- Real Inner Products
- Angles and Orthogonality
- Orthogonal and Orthonormal Bases
- Orthogonal Projections
Chapter 10: ORTHOGONAL
MATRICES
- Introduction
- Eigenvalues and Eigenvectors
- Orthonormal Diagonalization
Chapter 11: APPLICATIONS
OF REAL INNER PRODUCT SPACES
- Least Squares Approximation
- Quadratic Forms
- Real Fourier Series
Chapter 12: COMPLEX
VECTOR SPACES
- Complex Inner Products
- Unitary Matrices
- Unitary Diagonalization