MATH 270 Study Guide / MATH270 Study Guide: Athabasca University, Calgary

,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

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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