数学科学研究所
Insitute of Mathematical Science

线性代数II大纲

《线性代数II》教学大纲

一、课程基本信息

开课单位:

数学科学研究所

课程代码:

MATH1122

课程名称:

线性代数II

英文名称:

Linear     Algebra II

:

4

:

64

授课对象:

 

授课语言:

中英文

先修课程:

线性代数I

二、课程简介和教学目的

This is an advanced course of “Linear Algebra, I”, and we will take a more abstract view of points on the subjects in linear algebra. That is to say, we will begin with notions of vector spaces and linear maps. They are actually the abstract version of R^n and matrices. Then we will go through different types of vector spaces, like inner product spaces, real and complex vector spaces. Finally, we hope to end up with the studying of operators on the  these special vector spaces. This course is the preparation for more advanced courses like “Algebra” and “Functional Analysis”.

三、教学内容、教学方式和学时安排

(each     table is for 1 week, 4 periods )

Chapter        1

Vector        Spaces



1.1

Real and complex numbers



1.2

Definition of vector spaces



1.3

Subspaces



 

Chapter        2

Finite-dimensional        vector spaces



2.1

Span and linear independence



2.2

Bases



2.3

Dimension



 

Chapter        3

Linear        Maps



3.1

The vector spaces of Linear Maps



3.2

Null spaces and Ranges



3.3

Matrices



 

Chapter        4

More on        Linear Maps



4.1

Isomorphisms



4.2

Products and Quotients



4.3

Duality



 

Chapter        5

Polynomials        



5.1

Uniqueness of coefficients for Polynomials



5.2

The Division Algorithm



5.3

Factorization of Polynomials



 

Chapter        6

Invariant        Subspaces



6.1

Invariant subspaces



6.2

Eigenvectors and Eigenvalues



6.3

Polynomials applied to operators



 

Chapter        7

Upper-Triangluar        Matrices



7.1

Existence of eigenvalues



7.2

Upper-Triangluar       Matrices



7.3

Diagonal Matrices



 

Chapter        8

Inner        product spaces



8.1

Inner products and Norms



8.2

Orthonormal Bases 



8.3

Orthonormal complements



 

Chapter        9

Operators        on Inner product spaces



9.1

Self-adjoint operators



9.2

Normal operators



9.3

The Spectral Theorem



 

Chapter        10

Positive        Operators



10.1

Positive operators



10.2

Polar Decomposition



10.3

Singular Value Decomposition



 

Chapter        11

Operators        on complex vector spaces



11.1

Nilpotent operators



11.2

Generalized eignevectors



11.3

Decomposition of an operator



11.4




 

Chapter        12

More on        complex vector spaces



12.1

Square root of metrices



12.2

Block Matrices



12.3

Characteristic Polynomials



 

Chapter        13

Jordan        Forms



13.1

Minimal Polynomials



13.2

Jordan blocks



13.3

Jordan decomposition



 

Chapter        14

Operator        on Real Vector spaces



14.1

Complexification of a vector space



14.2

Complexification of an operator



 

Chapter        15

Complexification        continued



15.1

The minimal Polynomial of the complexification



15.2

Characteristic Polynomial of the complexification 



 

 

Chapter        16

Traces        and Determinant



16.1

Change of basis



16.2

Determinant of an operator



Final        Exam

The        17th. Week



 


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