Chapter 1 Basics of Linear Algebra 1.1 Basic operations of matrices 1.1.1 Addition for matrices 1.1.2 Scalar multiplication of matrix A 1.1.3 Matrix multiplication 1.1.4 Identity matrix 1.1.5 Transposition matrix and conjugate transpose of matrix A 1.1.6 Matrix inversion 1.1.7 Symmetries 1.2 Determinants 1.2.1 Determinant definition 1.2.2 Properties of determinants 1.2.3 Cofactor 1.2.4 Cramer's rule 1.3 Elementary operations 1.3.1 Elementary row transformation 1.3.2 Reduced echelon form 1.3.3 Rank of matrix 1.3.4 Solving equations by elementary transformation 1.4 Linear independence 1.5 Exercises Chapter 2 Linear Space 2.1 Set and map 2.2 Linear space 2.3 Basis, dimension and coordinates 2.4 Change of basis 2.5 Exercises Chapter 3 Normed Linear Space and Inner Product Space 3.1 Normed linear space and matrix norm 3.1.1 Normed linear space 3.1.2 Norm of matrix 3.2 Inner product spaces 3.2.1 Inner product 3.2.2 Representation of inner product 3.2.3 Orthogonality and Schmidt's orthogonalization method 3.3 Application of norm-preliminary matrix analysis 3.3.1 The limit of matrix sequence 3.3.2 Matrix series 3.3.3 Matrix power series 3.3.4 Differentiation and integration of matrices 3.4 Exercises Chapter 4 Linear Transformation 4.1 Linear transformation 4.2 Matrix of linear transformation 4.3 Eigenvalues and eigenvectors 4.4 Eigenvalues and eigenvectors for matrix 4.5 Exercises Chapter 5 Jordan Normal Form of Matrix and Matrix Function 5.1 Diagonalization 5.2 Jordan normal form of matrix A
5.3 Minimum polynomial 5.4 Matrix functions 5.4.1 Matrix function by infinite series 5.4.2 General definition and calculation of matrix function 5.4.3 Applications 5.5 Exercises Chapter 6 Applications of Matrix Theory in Linear Equations and Matrix Equations 6.1 Matrix factorization and application in linear equations 6.1.1 The LU factorization and applications 6.1.2 Applications in solving linear equations 6.2 Minus inverse and applications in compatible linear equations 6.3 Plus inverse of matrix and the minimal norm least square solutions of linear equations 6.3.1 Full rank factorization of matrix 6.3.2 Plus inverse of matrix 6.3.3 Minimal norm least square solution of linear equations 6.4 Kronecker product and applications in matrix equations 6.4.1 Definitions and properties of Kronecker product 6.4.2 Applications in matrix equations 6.5 Exercises References
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