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Preface
This module is developed to meet the specific needs of students who
may feel that they have deficiency in linear algebra and those students who
have completed an undergraduate course in linear algebra. Each chapter
begins with the learning objectives and pertinent definitions and theorems.
All the illustrative examples and answers to the self-assessment quiz are
fully explained to ensure complete understanding. The self-assessment quiz
can provide students with ample opportunity for practice and can help
them evaluate their progress.
This self-learning module aims to provide students enrolled in the
Open University of the Central Luzon State University a thorough
understanding of the fundamental concepts of linear algebra and matrix
theory through distance learning. It consists of seven chapters: Chapter 1
deals with matrix operations and their properties; Chapter 2 covers
solutions of systems of linear equations using the Gauss-Jordan elimination
method, determinants and its properties. Vectors in ℝ𝑛 , vector spaces and
spanning sets are presented in Chapter 3. Chapter 4 deals with linear
independence, basis and dimension of a matrix; Chapter 5 covers linear
transformations and matrices. Eigenvalues, eigenvectors and
diagonalization are discussed in Chapter 6 and inner product spaces, GramSchmidth orthogonalization process and diagonalization of symmetric
matrix are discussed in Chapter 7.
The author would like to thank those who have shared in the
successful completion of this module. Dr. Flor Amor B. Monta, Director of
Open University for the support, trust and confidence he gave to the
author; Dr. Ruben C. Sevilleja, President of Central Luzon State University,
for his continuous support and commitment to improve mathematics
education; and, above all, to the Almighty God, for His bountiful blessings
and graces.
The Author
Table of Contents
Preface
Chapter 1
Chapter 2
Chapter 3
Matrices Over a Field F
Introduction
Objectives
Definition of a Field
Definition of a Matrix
Matrix Operations
Properties of Matrix Addition
Properties of Matrix Multiplication
Properties of Scalar Multiplication
Transpose of a Matrix
Special Types of Square Matrices
1
1
1
2
3
7
9
11
12
14
Linear Equations and Matrices
Introduction
Objectives
Solutions of Systems of Linear Equations
Elementary Row Operations
The Gauss-Jordan Method
Homogeneous Systems
The Inverse of a Matrix
Linear Systems and Inverses
Determinants
Properties of Determinants
Minors and Cofactor Expansion
19
19
19
23
27
32
35
41
46
49
53
Vector Spaces Over a Field
Introduction
Objectives
Vectors in the Plane
Vector Operations
𝒏-Vectors
Vector Spaces and Subspaces
Linear Combinations and Spanning Sets
64
64
64
65
65
67
76
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Linear Independence
Introduction
Objectives
Definition of Linear Independence
Basis and Dimension
Rank of a Matrix
Linear Transformations and Matrices
Introduction
Objectives
Linear Transformations
The Kernel and Range of a Linear
Transformation
The Matrix of a Linear Transformation
Eigenvalues and Eigenvectors
Introduction
Objectives
Characteristic Polynomial
Hamilton-Cayley Theorem
Eigenvalues, Eigenvectors, and Eigenspaces
Diagonalization
Inner Product Spaces
Introduction
Objectives
Inner Product in ℝ𝑛
Orthonormal Bases in ℝ𝑛
Gram-Schmidt Orthogonalization
Process
Diagonalization of Symmetric Matrix
82
82
82
86
94
100
100
100
105
113
117
117
117
118
121
128
131
131
131
134
134
139