,Course Team
Course Professor: Gustavo Carrero, Ph.D.
Authors: Gustavo Carrero, Ph.D., and Arzu Sardarli, Ph.D.
Editor: Yasamine Coulter
Visual Communication Designer: Jingfen Zhang
Multimedia/Web Specialist: Audrey Krawec
Cover image:
Retrieved from:
https://commons.wikimedia.org/wiki/File:Mona Lisa eigenvector grid.png
By TreyGreer62 (Image: Mona Lisa-restored.jpg) [CC0], via Wikimedia Commons
Every effort has been taken to ensure that these materials comply with the
requirements of copyright clearances and appropriate credits. Athabasca University
will attempt to incorporate in future printings any corrections which are
communicated to it.
The inclusion of any material in this publication is strictly in accord with the
consents obtained and Athabasca University does not authorize or license any
further reproduction or use without the consent of the copyright holder.
c Athabasca University 2016
All rights reserved
Revision C7
,Contents
Introduction 1
1 Systems of Linear Equations and Matrices 3
1.1 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . 5
1.2 Gaussian and Gauss-Jordan Elimination . . . . . . . . . . . . . . . 7
1.3 Matrices, Their Operations and Their Algebraic Properties . . . . . 9
1.4 Applications of Linear Algebra: Ancient Applications, Traffic Flow
and Chemical Equations . . . . . . . . . . . . . . . . . . . . . . . 11
2 Inverse of a Matrix, Linear Systems and Special Forms of Matrices 13
2.1 Inverse of a Matrix and Its Properties . . . . . . . . . . . . . . . . . 15
−1
2.2 A Method for Finding the Inverse A . . . . . . . . . . . . . . . . 17
2.3 Linear Systems and the Inverse of a Matrix . . . . . . . . . . . . . 20
2.4 Special Forms of Matrices . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Applications of Linear Algebra to Economic Systems: Leontief
Economic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Determinant of a Matrix 25
3.1 Defining the Determinant with Minors and Cofactors . . . . . . . . 26
3.2 Determinants by Row Reduction; Properties of the Determinant . . 30
3.3 More Properties of the Determinant; Adding to the Equivalent
Statements Theorem for an Invertible Matrix . . . . . . . . . . . . . 31
3.4 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5 Applications of Linear Algebra to Geometry: Constructing Curves
and Surfaces Using Determinants . . . . . . . . . . . . . . . . . . . 34
4 Euclidean Vector Spaces: IR2 , IR3 and IRn 35
4.1 Vector Operations and Properties . . . . . . . . . . . . . . . . . . . 37
4.2 Lengths, Distances and Dot / Inner Product . . . . . . . . . . . . . 39
4.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4 Vector and Parametric Equations of Lines and Planes . . . . . . . . 42
4.5 Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.6 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . 45
4.7 Applications of Linear Algebra to Dynamical Systems: Markov
Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5 General Vector Spaces 49
5.1 Real Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
, 5.3 Linear Independence, Basis and Dimension . . . . . . . . . . . . . 55
5.4 Introduction to Linear Transformations: Basic Matrix
Transformations in IR2 and IR3 . . . . . . . . . . . . . . . . . . . . 57
5.5 Applications of Linear Algebra to Computer Graphics:
Transforming Images with Matrix Operators . . . . . . . . . . . . . 58
Course Professor: Gustavo Carrero, Ph.D.
Authors: Gustavo Carrero, Ph.D., and Arzu Sardarli, Ph.D.
Editor: Yasamine Coulter
Visual Communication Designer: Jingfen Zhang
Multimedia/Web Specialist: Audrey Krawec
Cover image:
Retrieved from:
https://commons.wikimedia.org/wiki/File:Mona Lisa eigenvector grid.png
By TreyGreer62 (Image: Mona Lisa-restored.jpg) [CC0], via Wikimedia Commons
Every effort has been taken to ensure that these materials comply with the
requirements of copyright clearances and appropriate credits. Athabasca University
will attempt to incorporate in future printings any corrections which are
communicated to it.
The inclusion of any material in this publication is strictly in accord with the
consents obtained and Athabasca University does not authorize or license any
further reproduction or use without the consent of the copyright holder.
c Athabasca University 2016
All rights reserved
Revision C7
,Contents
Introduction 1
1 Systems of Linear Equations and Matrices 3
1.1 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . 5
1.2 Gaussian and Gauss-Jordan Elimination . . . . . . . . . . . . . . . 7
1.3 Matrices, Their Operations and Their Algebraic Properties . . . . . 9
1.4 Applications of Linear Algebra: Ancient Applications, Traffic Flow
and Chemical Equations . . . . . . . . . . . . . . . . . . . . . . . 11
2 Inverse of a Matrix, Linear Systems and Special Forms of Matrices 13
2.1 Inverse of a Matrix and Its Properties . . . . . . . . . . . . . . . . . 15
−1
2.2 A Method for Finding the Inverse A . . . . . . . . . . . . . . . . 17
2.3 Linear Systems and the Inverse of a Matrix . . . . . . . . . . . . . 20
2.4 Special Forms of Matrices . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Applications of Linear Algebra to Economic Systems: Leontief
Economic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Determinant of a Matrix 25
3.1 Defining the Determinant with Minors and Cofactors . . . . . . . . 26
3.2 Determinants by Row Reduction; Properties of the Determinant . . 30
3.3 More Properties of the Determinant; Adding to the Equivalent
Statements Theorem for an Invertible Matrix . . . . . . . . . . . . . 31
3.4 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5 Applications of Linear Algebra to Geometry: Constructing Curves
and Surfaces Using Determinants . . . . . . . . . . . . . . . . . . . 34
4 Euclidean Vector Spaces: IR2 , IR3 and IRn 35
4.1 Vector Operations and Properties . . . . . . . . . . . . . . . . . . . 37
4.2 Lengths, Distances and Dot / Inner Product . . . . . . . . . . . . . 39
4.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4 Vector and Parametric Equations of Lines and Planes . . . . . . . . 42
4.5 Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.6 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . 45
4.7 Applications of Linear Algebra to Dynamical Systems: Markov
Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5 General Vector Spaces 49
5.1 Real Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
, 5.3 Linear Independence, Basis and Dimension . . . . . . . . . . . . . 55
5.4 Introduction to Linear Transformations: Basic Matrix
Transformations in IR2 and IR3 . . . . . . . . . . . . . . . . . . . . 57
5.5 Applications of Linear Algebra to Computer Graphics:
Transforming Images with Matrix Operators . . . . . . . . . . . . . 58
