Linear Algebra

by WWL Chen

This set of notes has been compiled over a period of some 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:

• Chapters 1, 2, 3 and 4.

MATH136 and MATH133:

• Chapters 5, 6 and 7.

MATH235:

• Chapters 8, 9, 10, 11 and 12.

To read the notes, click the chapters below for connection to the appropriate PDF files. You will need Adobe Acrobat Reader Version 4.0 or later.

The material is available free to all individuals, on the understanding that it is not to be used for financial gains, 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 (last uploaded on 28 February 2006)

• 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 (last uploaded on 10 April 2006)

• 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 (last uploaded on 19 April 2006)

• 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 (last uploaded on 4 May 2006)

• 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 (last uploaded on 24 February 2005)

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

Chapter 6: VECTOR SPACES ASSOCIATED WITH MATRICES (last uploaded on 24 February 2005)

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

Chapter 7: EIGENVALUES AND EIGENVECTORS (last uploaded on 24 February 2005)

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

Chapter 8: LINEAR TRANSFORMATIONS (last uploaded on 10 April 2006)

• 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 (last uploaded on 28 February 2006)

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

Chapter 10: ORTHOGONAL MATRICES (last uploaded on 28 February 2006)

• Introduction
• Eigenvalues and Eigenvectors
• Orthonormal Diagonalization

Chapter 11: APPLICATIONS OF REAL INNER PRODUCT SPACES (last uploaded on 28 February 2006)

• Least Squares Approximation
• Quadratic Forms
• Real Fourier Series

Chapter 12: COMPLEX VECTOR SPACES (last uploaded on 28 February 2006)

• Complex Inner Products
• Unitary Matrices
• Unitary Diagonalization