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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