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# Linear Algebra

535

Authors: Ubaidullah, Rajasegaran and Sireesha

ISBN: 9789391549893

Pages: 552

Binding: Softcover

Publisher: Yes Dee Publishing

SKU: 9789391549893 Category:

## Description

This text is designed to support students irrespective of their prior knowledge in Linear Algebra. Every chapter begins with an introduction of the usage in various fields as a teaser followed by clear explanation of the concepts with statements of pertinent definitions and theorems. It focuses on geometric explanations in selected topics to illustrate that Linear Algebra is Analytic Geometry of n dimensions.

This book contains ample examples concerning linear equations, computations with matrices and determinants, vector spaces, eigenvalues, eigenvectors, diagonalization of matrices and orthogonal vectors. Summary of key terms and formulae at the end of each chapter brings greater focus to the material that must be learned.

Key Features:

• Pedagogical Features
• Geometric Emphasis
• Technology Reference
• Self- Check, Section Practices and Supplementary Exercises

Weight 0.6 kg 10 × 7 × 3 cm

Chapter 1  Preliminaries

1.1 Introduction

1.2 Properties of Real Numbers

1.3 Complex Number

Examples 1.1 to 1.6 2

1.4 Set

Examples 1.7 to 1.13

1.5 Relation and Function

Examples 1.14 to 1.16

1.6 Polynomial

Examples 1.17 to 1.21

Chapter 2  Linear Equation

2.1 Introduction

2.1.1 Linear Equation

2.1.2 System of n Linear Equations (Linear System)

Examples 2.1 to 2.23

Technology reference

Self-check

Section practices

2.2 Solving Two Variable System

2.2.1 Two Variable System by Elimination Method

Example 2.4

2.2.2 Two Variable System by Substitution Method

Example 2.5 26

2.2.3 Two Variable System by Graphical Interpretation

Examples 2.6 to 2.8

Technology reference

Self-check

Section practices

2.3 Row-Echelon Form, Solving Multi Variable System

2.3.1 Row-Echelon Form

2.3.2 Solving Three Variable System

Examples 2.9 to 2.15

Technology reference

Self-check

Section practices

2.4 Reduced Row-Echelon Form, Solving Multi Variable System

2.4.1 Transforming a 2 × 2 Matrix into Reduced Row-Echelon Form

Example 2.16

2.4.2 Transforming a 3 × 3 Matrix into Reduced Row-Echelon Form

Example 2.17

2.4.3 Transforming a 2 × 3 Matrix into Reduced Row-Echelon Form

Example 2.18

2.4.4 Transforming a 3 × 2 Matrix into Reduced Row-Echelon Form

Example 2.19

2.4.5 Transforming a 4 × 4 Matrix into Reduced Row-Echelon Form

Example 2.20

2.4.6 Gauss-Jordan Method by Row Reduction

Examples 2.21 to 2.22

Technology reference

Self-check

Section practices

2.5 Solving Homogeneous System

2.5.1 Homogenous Equations with Trivial Solution

Examples 2.23 to 2.24

2.5.2 Homogeneous Systems with Many Solutions or Non-trivial

Solutions

Examples 2.25 to 2.26

Technology reference

Self-check

Section practices

2.6 Vector Equations, Span and Linear Independence

2.6.1 Vector Equation

Examples 2.27 to 2.28

2.6.2 Span of a Set of Vectors

Examples 2.29 to 2.30

2.6.3 Homogeneous Linear System and Spans

Example 2.31

2.6.4 Linear Independence and Linear Dependence

Examples 2.32 to 2.33

Technology reference

Self-check

Section practices

2.7 Linear Transformation and Matrix Transformation

Examples 2.34 to 2.39

Self-check

Section practices

Supplementary exercises

Summary of key terms and formulae

Chapter 3 Matrix Algebra

3.1 Introduction

3.1.1 Dimension of a Matrix [Order of a Matrix]

Examples 3.1 to 3.5

3.1.2 Some Special Matrices

Examples 3.6 to 3.20

Technology reference

Self-check

Section practices

3.2 Matrix Operations

Examples 3.21 to 3.32

3.2.2 Scalar Multiplication

Examples 3.33 to 3.38

3.2.3 Matrix Multiplication [Product of Two Matrices]

Examples 3.39 to 3.52

3.2.4 Powers of a Matrix

Examples 3.53 to 3.5

Technology reference

Self-check

Section practices

3.3 Inverse of  a Matrix

3.3.1 Inverse of a Matrix [Inverse of a Non-Singular Matrix]

Examples 3.55 to 3.62

3.3.2 Determinant (3rd Order)

Examples 3.63 to 3.64

3.3.3 Singular Matrix [Non-Invertible Matrix]

Examples 3.65 to 3.66

3.3.4 Non-Singular Matrix [Invertible Matrix]

Examples 3.67 to 3.68

Examples 3.69 to 3.70

3.3.6 Inverse of the Matrix (3 × 3) by Adjoint Method or Co-factor

Expansion

Examples 3.71 to 3.72

3.3.7 Inverse of the Matrix (3×3) using an Algorithm [Gauss-Jordan

Reduction]

Examples 3.73 to 3.74

Technology reference

Self-check

Section practices

3.4 Invertible Matrix and its Applications

3.4.1 Invertible Matrix Theorem

Examples 3.75 to 3.76

Self-check

Section practices

3.5 Inverse and System of Equations

3.5.1 Matrix Inversion Method to Solve Linear System

Examples 3.77 to 3.78

Self-check

Section practices

Supplementary exercises

Summary of key terms and formulae

Chapter 4  Determinants

4.1 Introduction

4.2 Determinant of a Matrix

4.2.1 Determinant of 2 × 2 Matrix (2nd Order Determinant)

Examples 4.1 to 4.2

4.2.2 Minor of an Entry and Co-factor of an Entry

Examples 4.3 to 4.7

4.2.3 Determinant of 3 × 3Matrix(3rdOrder)

Examples 4.8 to 4.9

4.2.4 Rule of SARRUS − Alternative Form for a Determinant of

3rdOrder

Example 4.10

Technology reference

Self-check

Section practices

4.3 Properties of Determinant

4.3.1 Property 1: Zero Property

Examples 4.11 to 4.19

4.3.2 Property 2: Triangular Property

Examples 4.20 to 4.21

4.3.3 Property 3: Reflection Property

Examples 4.22 to 4.23

4.3.4 Property 4: Scalar Multiple Property

Examples 4.24 to 4.26

4.3.5 Property 5: Switching Property

Example 4.27

4.3.6 Property 6: Sum Property

Examples 4.28 to 4.29

4.3.7 Property 7: Invariance Property

Example 4.30

4.3.8 Property 8: Product of Determinant Property

Example 4.31

4.3.9 Property 9: Determinant of the Inverse Property

Examples 4.32 to 4.33

4.3.10 Property 10: Adjoint of a Matrix Property

Example 4.34

4.3.11 Evaluating Determinant by Row Reduction

Examples 4.35 to 4.36

4.3.12 Vandermonde Determinant by Row Reduction

Example 4.37

Technology reference

Self-check

Section practices

4.4 Cramer’s Rule and System of Linear Equations

4.4.1 Cramer’s Rule and System of Linear Equations

4.4.2 Cramer’s Rule for Non-Homogeneous System of Linear

Equations of n Unknowns

4.4.3 Cramer’s Rule for Non-Homogeneous System of Linear

Equations of Two Unknowns

Example 4.38

4.4.4 Cramer’s Rule for Non-Homogeneous System of Linear

Equations of Three Unknowns

Example 4.39

4.4.5 Cramer’s Rule for Non-Homogeneous System of Linear

Equations of Four Unknowns

Example 4.40

4.4.6 Cramer’s Rule for Homogeneous System of Linear Equations
of n Unknowns

Example 4.41

Technology reference

Self-check

Section practices

4.5 Applications of Determinant

4.5.1 Cross Product of Three-Dimensional Vectors using Determinant

Example 4.42

4.5.2 Properties of Cross Product of Three-Dimensional Vectors

Examples 4.43 to 4.48

4.5.3 Two Dimension – Area of the Parallelogram using Determinant

Examples 4.49 to 4.56

Technology reference

Self-check

Section practices

Supplementary exercises

Summary of key terms and formulae

Chapter 5 Vector Spaces

5.1 Introduction

5.1.1 Set Notation

5.1.2 Vector Space Notation

5.2 Vector Spaces and Subspaces

5.2.1 Vector Space

5.2.2 Determining Vector Space

Examples 5.1 to 5.2

5.2.3 Vector Space of m × n Matrices

Examples 5.3 to 5.5

5.2.4 Vector Space with Polynomials

Example 5.6

5.2.5 Additional Properties of Vector Spaces

5.2.6 Subspace

5.2.7 Determining Subspace

Examples 5.7 to 5.9

Technology reference

Self-check

Section practices

5.3 Null Space, Row Space and Column Space

5.3.1 Null Space of a Matrix

5.3.2 Finding Null Space of a Matrix

Examples 5.10 to 5.13

5.3.3 Row Space of a Matrix

Examples 5.14 to 5.15

5.3.4 Column Space of a Matrix

Examples 5.16 to 5.17

Technology reference

Self-check

Section practices

5.4 Basis, Dimension, Rank and Nullity of a Matrix

Examples 5.18 to 5.23

Technology reference

Self-check

Section practices

5.5 Linear Combination, Span of Vectors, Spanning Set

Examples 5.24 to 5.33

Technology reference

Self-check

Section practices

5.6 Linear Independence and Wronskian

Examples 5.34 to 5.44

Technology reference

Self-check

Section practices

5.7 Coordinate Vector and Change-of-Basis

Examples 5.45 to 5.50

Self-check

Section practices

Supplementary exercises

Summary of key terms and formulae

Chapter 6 Eigenvalues and Eigenvectors

6.1 Introduction

6.2 Eigenvalues and Eigenvectors

6.2.1 Computation of Eigenvalues of a 2 × 2Matrix 381

Example 6.1

6.2.2 Computation of Eigenvectors of a 2 × 2Matrix 383

Examples 6.2 to 6.6

6.2.3 Computation of Eigenvalues of a 3 × 3Matrix 389

Example 6.7

6.2.4 Computation of Eigenvectors of a 3 × 3Matrix 393

Examples 6.8 to 6.10

Technology reference

Self-check

Section practices

6.3 Properties of Eigenvalues

Examples 6.11 to 6.21

Technology reference

Self-check

Section practices

6.4 Cayley-Hamilton Theorem

6.4.1 Finding the Inverse of a Non-Singular Matrix using

Cayley-Hamilton Theorem

Example 6.22

6.4.2 Finding the Higher Positive Power of a Matrix A using

Cayley-Hamilton Theorem

Examples 6.23 to 6.25

Technology reference

Self-check

Section practices

6.5 Complex Eigenvalues and Complex Eigenvectors

6.5.1 Computation of Complex Eigenvalues and Complex Eigenvectors

of 2 × 2Matrix

Example 6.26

6.5.2 Computation of Complex Eigenvalues and Complex Eigen

Vectors of 3 × 3Matrix

Example 6.27

Technology reference

Self-check

Section practices

6.6 Diagonalization of a Matrix

6.6.1 Diagonalization of a Square Matrix

6.6.2 Diagonalization of a 2 × 2Matrix

Example 6.28

6.6.3 Diagonalization of a 3 × 3SymmetricMatrix

Examples 6.29 to 6.30

6.6.4 Powers of a Matrix using Diagonalization

Example 6.31

Technology reference

Self-check

Section practices

6.7 Basis of Eigenspace of a Matrix

6.7.1 Eigenspace of a 2 × 2Matrix

Example 6.32

6.7.2 Eigenspace of a 3 × 3Matrix

Example 6.33

6.7.3 Eigenspace of a 4 × 4Matrix

Example 6.34

Self-check

Section practices

Supplementary exercises

Summary of key terms and formulae

Chapter 7 Orthogonality

7.1 Introduction

7.2 Vector Representations and Operations

7.2.1 Vector Representations and Operations

7.2.2 Types of Vectors

Examples 7.1 to 7.6

7.2.3 Vector Operations and Properties

Example 7.7

7.2.4 Properties of Addition of Vectors and Multiplication of a Vector

By a Scalar

Examples 7.8 to 7.12

7.2.5 Rectangular Components in 2D

7.2.6 Orthogonal System of Unit Vectors i, j and k

Example 7.13

7.2.7 Norm and Distance in Rn

Examples 7.14 to 7.19

Technology reference

Self-check

Section practices

7.3 Inner Product Space

7.3.1 Inner Product or Dot Product

Examples 7.20 to 7.27

7.3.2 Orthogonal Vectors − Orthogonality

Examples 7.28 to 7.32

Technology reference

Self-check

Section practices

7.4 Orthogonal Complement and Orthogonal Projection

7.4.1 Orthogonal Complement

Examples 7.33 to 7.34

7.4.2 Orthogonal Projection Onto a Line

Examples 7.35 to 7.38

Technology reference

Self-check

Section practices

7.5 Gram-Schmidt Orthogonalization Process

7.5.1 Gram-Schmidt Orthogonalization Process

Examples 7.39 to 7.41

7.5.2 Orthonormal Basis

Example 7.42

Technology reference

Self-check

Section practices

7.6 Orthogonal Matrices

7.6.1 Orthogonal Matrices

Examples 7.43 to 7.47

7.6.2 Orthogonal Diagonalization Matrices

Example 7.48

Self-check

Section practices

Supplementary exercises

Summary of key terms and formulae

Bibliography

Index

1. M. Mohamed Ubaidullah is Senior Lecturer at Asia Pacific University of Technology and Innovation, Kuala Lumpur, Malaysia. He has over twenty six years of teaching in academia and four years in industry. He has taught thirteen different Mathematics modules across all levels at the University. He is the recipient of the Vice Chancellors’ commendation for Teaching Excellence in Asia Pacific University for the years 2014, 2015 and 2018.

Dr. Rajasegeran Ramasamy is Associate Professor and Head at Asia Pacific University of Technology and Innovation, Kuala Lumpur, Malaysia. He has more than twenty years of experience in academia and industry. He has taught various mathematics courses including linear algebra and is also involved in mathematics modelling in industry, especially in optimization the resources.

Prathigadapa Sireesha is Senior Lecturer at Asia Pacific University of Technology and Innovation, Kuala Lumpur, Malaysia. She has more than fifteen years of teaching experience in various colleges and universities. She is the recipient of the Microsoft Innovative Educator Expert conferred by Microsoft Education for the year 2021-2022.

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