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LINEAR ALGEBRA

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Table of Contents
Chapter 1 Matrices and Determinants
1.1 Matrices
1.2 Matrix Arithmetic
1.2.1 Equality
1.2.2 Scalar Multiplication
1.2.3 Matrix Addition
1.2.4 Matrix Multiplication
1.2.5 Transpose of a Matrix
1.3 Determinants of Square Matrices
1.3.1 Second Order Determinant
1.3.2 n-th Order Determinant
1.3.3 Properties of Determinants
1.3.4 Evaluation of Determinants
1.3.5 Laplace's Theorem
1.4 Block Matrices
1.4.1 The Concept of Block Matrices
1.4.2 Evaluation of Block Matrices
1.5 Invertible Matrices
1.6 Elementary Matrices
1.6.1 Elementary Operations of Matrices
1.6.2 Elementary Matrices
1.6.3 Use Elementary Operations to Get the Inverse Matrix
1.7 Rank of Matrices
1.8 Exercises

Chapter 2 Systems of Linear Equations
2.1 Systems of Linear Equations
2.2 Vectors
2.3 Linear Independence
2.3.1 Linear Combination
2.3.2 Linear Dependence and Linear Independence
2.4 Maximally Linearly Independent Vector Group
2.4.1 Equivalent Vector Sets
2.4.2 Maximally Linearly Independent Group
2.4.3 The Relationship Between Rank of Matrices and Rank of Vector Sets
2.5 Vector Space
2.6 General Solutions of Linear Systems
2.6.1 General Solutions of Homogenous Linear Systems
2.6.2 General Solutions of Non-homogenous Linear Systems
2.7 Exercises

Chapter 3 Eigenvalues and Eigenveetors
3.1 Eigenvalues and Eigenvectors
3.1.1 Definition of Eigenvalues and Eigenvectors
3.1.2 Properties of Eigenvalues and Eigenvectors
3.2 Diagonalization o{ Square Matrices
3.2.1 Similar Matrix
3.2.2 Diagonalization of Square Matrices
3.3 Orthonormal Basis
3.3.1 Inner Product of Vectors
3.3.2 Orthogonal Set and Basis
3.3.3 Gram-Schmidt Orthogonalization Process
3.3.4 Orthogonal Matrix
3.4 Diagonalization of Real Symmetric Matrices
3.4.1 Properties of Eigenvalues of Real Symmetric Matrices
3.5 Exercises

Chapter 4 Quadratic Form
4.1 Real Quadratic Form and Its Matrix
4.2 Diagonal Form of Quadratic Form
4.3 Diagonal Form of Real Quadratic Form
4.3.1 Changing Quadratic Form into Diagonal Form by Orthogonal Transformation
4.3.2 Changing Quadratic Form into Diagonal Form by the Method of Completing the Square
4.4 Canonical Form of Real Quadratic Form
4.5 Positive Definite Quadratic Form and Matrices
4.6 Exercises
References
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LINEAR ALGEBRA
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