## Table of Contents

**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

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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

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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

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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

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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

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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

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2.7 Linear Transformation and Matrix Transformation

Examples 2.34 to 2.39

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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

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3.2 Matrix Operations

3.2.1 Matrix Addition and Subtraction

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

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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

3.3.5 Adjoint of a Matrix

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

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3.4 Invertible Matrix and its Applications

3.4.1 Invertible Matrix Theorem

Examples 3.75 to 3.76

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3.5 Inverse and System of Equations

3.5.1 Matrix Inversion Method to Solve Linear System

Examples 3.77 to 3.78

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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

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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

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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

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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

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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

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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

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5.4 Basis, Dimension, Rank and Nullity of a Matrix

Examples 5.18 to 5.23

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5.5 Linear Combination, Span of Vectors, Spanning Set

Examples 5.24 to 5.33

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5.6 Linear Independence and Wronskian

Examples 5.34 to 5.44

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5.7 Coordinate Vector and Change-of-Basis

Examples 5.45 to 5.50

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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

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6.3 Properties of Eigenvalues

Examples 6.11 to 6.21

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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

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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

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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

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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

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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

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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

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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

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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

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7.6 Orthogonal Matrices

7.6.1 Orthogonal Matrices

Examples 7.43 to 7.47

7.6.2 Orthogonal Diagonalization Matrices

Example 7.48

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Supplementary exercises

Summary of key terms and formulae

Bibliography

Index

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