Linear Algebra

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Table of Contents
Chapter 1 Determinant(1)
1.1 Definition of Determinant(1)
1.1.1 Determinant arising from the solution of linear system(1)
1.1.2 The definition of determinant of order n(5)
1.1.3 Determine the sign of each term in a determinant (8)
Exercise 1.1(10)
1.2 Basic Properties of Determinant and Its Applications(12)
1.2.1 Basic properties of determinant(12)
1.2.2 Applications of basic properties of determinant(15)
Exercise 1.2(19)
1.3 Expansion of Determinant (21)
1.3.1 Expanding a determinant using one row (column)(21)
1.3.2 Expanding a determinant along k rows (columns)(27)
Exercise 1.3(29)
1.4 Cramer’s Rule(30)
Exercise 1.4(36)

Chapter 2 Matrix(38)
2.1 Matrix Operations(38)
2.1.1 The concept of matrices(38)
2.1.2 Matrix Operations(41)
Exercise 2.1(58)
2.2 Some Special Matrices(60)
Exercise 2.2(64)
2.3 Partitioned Matrices(66)
Exercise 2.3(72)
2.4 The Inverse of Matrix(73)
2.4.1 Finding the inverse of an n×n matrix(73)
2.4.2 Application to economics(81)
2.4.3 Properties of inverse matrix (83)
2.4.4 The adjoint matrix A (or adjA) of A(86)
2.4.5 The inverse of block matrix(89)
Exercise 2.4(91)
2.5 Elementary Operations and Elementary Matrices(94)
2.5.1 Definitions and properties (94)
2.5.2 Application of elementary operations and elementary matrices(100)
2.6 Rank of Matrix(103)
2.6.1 Concept of rank of a matrix(104)
2.6.2 Find the rank of matrix(107)
Exercise 2.6(109)

Chapter 3 Solving Linear System by Gaussian Elimination Method(110)
3.1 Solving Nonhomogeneous Linear System by Gaussian Elimination Method(110)
3.2 Solving Homogeneous Linear Systems by Gaussian Elimination Method(128)
Exercise 3(131)

Chapter 4 Vectors(134)
4.1 Vectors and its Linear Operations(134)
4.1.1 Vectors(134)
4.1.2 Linear operations of vectors(136)
4.1.3 A linear combination of vectors (137)
Exercise 4.1(143)
4.2 Linear Dependence of a Set of Vectors (143)
Exercise 4.2(155)
4.3 Rank of a Set of Vectors(156)
4.3.1 A maximal independent subset of a set of vectors(156)
4.3.2 Rank of a set of vectors(159)
Exercise 4.3(163)

Chapter 5 Structure of Solutions of a System(165)
5.1 Structure of Solutions of a System of Homogeneous Linear Equations (165)
5.1.1 Properties of solutions of a system of homogeneous linear equations(165)
5.1.2 A system of fundamental solutions (166)
5.1.3 General solution of homogeneous system(171)
5.1.4 Solutions of system of equations with given solutions of the system(173)
Exercise 5.1(176)
5.2 Structure of Solutions of a System of Nonhomogeneous Linear Equations(178)
5.2.1 Properties of solutions(178)
5.2.2 General solution of nonhomogeneous equations (179)
5.2.3 The simple and convenient method of finding the system of fundamental solutions and particular solution(183)
Exercise 5.2(189)

Chapter 6 Eigenvalues and Eigenvectors of Matrices(191)
6.1 Find the Eigenvalue and Eigenvector of Matrix(191)
Exercise 6.1(197)
6.2 The Proof of Problems Related with Eigenvalues and Eigenvectors(198)
Exercise 6.2(199)
6.3 Diagonalization(200)
6.3.1 Criterion of diagonalization(200)
6.3.2 Application of diagonalization(209)
Exercise 6.3(210)
6.4 The Properties of Similar Matrices(211)
Exercise 6.4(216)
6.5 Real Symmetric Matrices(218)
6.5.1 Scalar product of two vectors and its basis properties(218)
6.5.2 Orthogonal vector set(220)
6.5.3 Orthogonal matrix and its properties(223)
6.5.4 Properties of real symmetric matrix(225)
Exercise 6.5(229)

Chapter 7 Quadratic Forms (231)
7.1 Quadratic Forms and Their Standard Forms(231)
Exercise 7.1(236)
7.2 Classification of Quadratic Forms and Positive Definite Quadratic(Positive Definite Matrix)(237)
7.2.1 Classification of Quadratic Form(237)
7.2.2 Criterion of a positive definite matrix(239)
Exercise 7.2(241)
7.3 Criterion of Congruent Matrices(242)
Exercise 7.3(245)

Answers to Exercises(246)

Appendix Index(266)
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