线性代数与计算

1 Preliminaries
1.1 Simple introduction to algebra
1.2 Set, map and number field
1.3 Polynomials and the algebraic system
1.3.1 Polynomials and polynomial equality
1.3.2 Operations of polynomials
1.4 Linear relations, bases and roots of polynomials
1.4.1 Linear relations and basis for polynomials
1.4.2 Roots of polynomials and their solutions
1.5 Computation and experiments
2 Systems of Equations and Matrices
2.1 Systems of linear equations
2.2 Solution of general systems and row echelon form
2.2.1 Solutions to general systems of linear equations
2.2.2 Row echelon form and Gaussian elimination
2.3 Matrix arithmetic
2.3.1 Equality, scalar multiplication, addition and subtraction
2.3.2 Representation of linear systems and consistency theorem
2.3.3 Matrix multiplication and transpose
2.4 Matrix algebra
2.4.1 Algebraic rules for matrix operations
2.4.2 Identity matrix and matrix inversion
2.5 Elementary matrices and their applications
2.5.1 Elementary matrices and their properties
2.5.2 The inverse of a matrix and LU factorization
2.6 Partitioned matrices
2.7 Computation and experiments
3 Determinants
3.1 Determinants of square matrices
3.2 Properties of determinants
3.3 Applications of determinants
3.3.1 Computation of inverses of matrices by determinants
3.3.2 Cramer's Rule
3.3.3 Computation of the area of a triangle
3.4 Computation and experiments
4 Vector Spaces
4.1 Definition of vector space and examples
4.1.1 Definitions of Rn and Rmxn
4.1.2 Definition of vector space
4.2 Subspace and spanning set
4.3 Linear independence, basis and dimension
4.3.1 Linear independence
4.3.2 Basis and dimensions of vector space
4.4 Row space and column space
4.4.1 Structures of solution sets for systems of equations
4.4.2 Row or column space and consistency theorem
4.5 Linear transformations
4.5.1 Linear transformations
4.5.2 Matrix representations of linear transformations
4.6 Computation and experiments
5 Inner Product Spaces and Eigenvalues
5.1 The inner products in R2 and R3
5.2 Inner product and orthogonality in general Rn
5.3 General inner product spaces and orthogonal matrices
5.4 The Gram-Schmidt orthogonalization process
5.5 Eigenvalues and eigenvectors
5.6 Quadratic forms and positive definite matrices
5.7 Computation and experiments
6 Numerical Algebra
6.1 Floating-point representation and round-off errors
6.2 Partial pivoting and condition number
6.2.1 Partial pivoting
6.2.2 Matrix norms and condition numbers
6.3 Iterative methods for systems of equations
6.3.1 Jacobi method
6.3.2 Gauss-Seidel method
6.3.3 Successive over-relaxation
6.4 Solving nonlinear equations
6.4.1 The bisection method
6.4.2 The fixed-point iteration method
6.4.3 Newton's method
6.5 Computation and experiments
7 Interpolation of Polynomials and Applications
7.1 Construction of polynomials by solving systems
7.2 Lagrange interpolation
7.3 Newton's divided difference formula
7.4 Numerical differentiation and integration
7.4.1 Numerical differentiation
7.4.2 Numerical integration
7.5 Computation and experiments
Bibliography