Chapter 1 Determinant 1.1 Definition of Determinant 1.1.1 Determinant arising from the solution of linear system 1.1.2 The definition of determinant of order n 1.1.3 Determine the sign of each term in a determinant Exercise 1.1 1.2 Basic Properties of Determinant and Its Applications 1.2.1 Basic properties of determinant 1.2.2 Applications of basic properties of determinant Exercise 1.2 1.3 Expansion of Determinant 1.3.1 Expanding a determinant using one row (column) 1.3.2 Expanding a determinant along k rows (columns) Exercise 1.3 1.4 Cramer』s Rule Exercise 1.4 Chapter 2 Matrix 2.1 Matrix Operations 2.1.1 The concept of matrices 2.1.2 Matrix Operations Exercise 2.1 2.2 Some Special Matrices Exercise 2.2 2.3 Partitioned Matrices Exercise 2.3 2.4 The Inverse of Matrix 2.4.1 Finding the inverse of an n×n matrix 2.4.2 Application to economics 2.4.3 Properties of inverse matrix 2.4.4 The adjoint matrix A? (or adjA) of A 2.4.5 The inverse of block matrix Exercise 2.4 2.5 Elementary Operations and Elementary Matrices 2.5.1 Definitions and properties 2.5.2 Application of elementary operations and elementary matrices Exercise2.5 2.6 Rank of Matrix 2.6.1 Concept of rank of a matrix 2.6.2 Find the rank of matrix Exercise 2.6 Chapter 3 Solving Linear System by Gaussian Elimination Method 3.1 Solving Nonhomogeneous Linear System by Gaussian Elimination Method 3.2 Solving Homogeneous Linear Systems by Gaussian Elimination Method Exercise 3 Chapter 4 Vectors 4.1 Vectors and its Linear Operations 4.1.1 Vectors 4.1.2 Linear operations of vectors 4.1.3 A linear combination of vectors Exercise 4.1
4.2 Linear Dependence of a Set of Vectors Exercise 4.2 4.3 Rank of a Set of Vectors 4.3.1 A maximal independent subset of a set of vectors 4.3.2 Rank of a set of vectors Exercise 4.3 Chapter 5 Structure of Solutions of a System 5.1 Structure of Solutions of a System of Homogeneous Linear Equations 5.1.1 Properties of solutions of a system of homogeneous linear equations 5.1.2 A system of fundamental solutions 5.1.3 General solution of homogeneous system 5.1.4 Solutions of system of equations with given solutions of the system Exercise 5.1 5.2 Structure of Solutions of a System of Nonhomogeneous Linear Equations 5.2.1 Properties of solutions 5.2.2 General solution of nonhomogeneous equations 5.2.3 The simple and convenient method of finding the system of fundamental solutions and particular solution Exercise 5.2 Chapter 6 Eigenvalues and Eigenvectors of Matrices 6.1 Find the Eigenvalue and Eigenvector of Matrix Exercise 6.1 6.2 The Proof of Problems Related with Eigenvalues and Eigenvectors Exercise 6.2 6.3 Diagonalization 6.3.1 Criterion of diagonalization 6.3.2 Application of diagonalization Exercise 6.3 6.4 The Properties of Similar Matrices Exercise 6.4 6.5 Real Symmetric Matrices 6.5.1 Scalar product of two vectors and its basis properties 6.5.2 Orthogonal vector set 6.5.3 Orthogonal matrix and its properties 6.5.4 Properties of real symmetric matrix Exercise 6.5 Chapter 7 Quadratic Forms 7.1 Quadratic Forms and Their Standard Forms Exercise 7.1 7.2 Classification of Quadratic Forms and Positive Definite Quadratic(Positive Definite Matrix) 7.2.1 Classification of Quadratic Form 7.2.2 Criterion of a positive definite matrix Exercise 7.2 7.3 Criterion of Congruent Matrices Exercise 7.3 Answers to Exercises Appendix Index
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