代数与几何

课程编码：08N1120220 总 学 时： 60 学 分： 3.5

先修课程：高中数学 授课教师：孟晨辉 教 材：《线性代数与空间解析几何（第三版）》， 高等教育出版社，郑宝东主编 课程简介：

代数与几何是高等学校功课各专业中十分重要的自然科学基础。其中的线性代数部分主要运用代数方法研究具有线性关系的数学对象，建立相应的理论体系，它具有很强的逻辑性与抽象性；其中的空间解析几何部分主要通过坐标系，建立空间几何图形与方程之间的关系，利用代数理论研究空间几何图形的性质。空间解析几何为线性代数提供背景与示例，线性代数与空间解析几何作为一门课程的两个组成部分，互相渗透，互相支持。

本课程系统的介绍线性代数与空间解析几何的基本理论与方法，把线性代数与空间解析几何的求解有机结合。内容包括行列式的定义、计算及其性质；矩阵的代数运算、分块矩阵、矩阵的求秩；向量代数，向量坐标，并在其中讨论几何问题、平面与直线的方程及其相互位置关系；n维向量、向量之间的线性相关性与线性无关性；线性方程组间的结构理论；线性变换与矩阵间的联系；特征值、特征向量、相似变换，矩阵可对角线化的条件和方法；二次型理论、二次型简化、二次曲面及其分类等。 评分标准：作业——20%

期中考试——20% 期末考试——60% 教学大纲： 一、n阶行列式

1.1 n阶行列式的概念 1.2 行列式的性质 1.3 行列式的展开订立 1.4 Crammer法则 二、矩阵

2.1 矩阵的概念 2.2 矩阵的运算 2.3 可逆矩阵

2.4 矩阵的初等变换 2.5 矩阵的秩 2.6 初等矩阵

2.7 分块矩阵的概念及其运算 2.8 分块矩阵的初等变换 三、几何向量 四、n维向量

4.1 n维向量的概念及其线性运算 4.2 向量组线性相关与线性无关 4.3 向量组的秩 4.4 向量空间 4.5 欧式空间

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五、线性方程组

5.1 线性方程组有解的充要条件 5.2 线性方程组解的结构

5.3 利用矩阵的初等行变化解线性方程组 5.4 线性方程组的几何应用 六、特征值、特征向量及相似矩阵

6.1 特征值与特征向量 6.2 相似矩阵 6.3 应用举例

七、线性空间与线性变换

7.1 线性空间的概念

7.2 线性空间的基地、维数与坐标 7.3 线性变换

八、二次型与二次曲面

8.1 实二次型

8.2 化实二次型为标准性 8.3 正定实二次型

8.4 空间中的曲面与曲线 8.5 二次曲面

Linear Algebra and Analytic Geometry

Course Code: 08N1120220 Hours: 60 Credits: 3.5

Prerequisite Course: College-entry level algebra and geometry Instructor: Chenhui Meng

Textbook:Baodong Zheng, Linear algebra and space analytic geometry, Higher Education Press Course Description

Linear Algebra and Analytic Geometry is one of the significant fundamental courses of nature science. The Linear Algebra mainly applies the algebra method to study the mathematic subjects that have the linear relation and establish corresponding theatrical system. The part of Linear Algebra introduces Basic concepts and techniques of linear algebra; includes systems of linear equations, matrices, determinants, vectors in n-space, and eigenvectors, together with selected applications, such as Markov processes, linear programming, economic models, least squares, and population growth.

The Analytic Geometry mainly applies coordinate systems to build the relationship between shapes and equations; and use algebra theory to study the characters of space geometry. The Analytic Geometry provides the background and examples to Linear Algebra. As two parts of the course, the linear algebra and analytic geometry they permeate each other. In addition, this course introduces techniques and applications of ordinary differential equations, including Fourier series and boundary value problems, linear systems of differential equations, and an introduction to partial differential equations. It is designed for engineering majors and other who require a working knowledge of differential equations. Grading: Homework--------------20% Midterm exam----------20%

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Final exam ---------------60% Syllabus:

Ⅰ n order determinant 1.1 Basic concepts

1.2 Determinant properties

1.3 Determinant expansion theorem 1.4 Cramer rules Ⅱ matrix

2.1 Basic concepts

2.2 Matrix manipulation 2.3 Invertible matrix

2.4 Elementary transformation of matrices 2.5 Rank of matrix 2.6 Elementary matrix 2.7 Block matrix

2.8 Elementary transformation of block matrices

Ⅲ Geometrical vector Ⅳ n dimensional vector 4.1 Concepts

4.2 Linear dependence and linear independence 4.3 Vector set rank 4.4 Vector space 4.5 European space

Ⅴ Linear system of equations

5.1 Necessary and sufficient condition for system of linear equations with solution 5.2 System of linear equations’ solution structure

5.3 Solving system of linear equations using matrix method 5.4 Geometric application for linear system of equations Ⅵ Eigenvalue, eigenvector, and similar matrix 6.1 Eigenvalue and eigenvector 6.2 Similar matrix 6.3 Examples

Ⅶ Linear space and linear transformation 7.1 Basic concepts

7.2 Basis, dimensionality, and coordinates 7.3 Linear transformation

Ⅷ Quadric form and Quadric surface 8.1 Real quadric form

8.2 Convert real quadric form into normal forms 8.3 Positive definite real quadric form 8.4 Surface and curve in space 8.5 Quadric surface

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工科数学分析

课程编码：08N1120211 08N1120212 总 学 时：90+90 学 分：5.5+5.5 先修课程：高中数学 授课教师：白红 教 材：《工科数学分析（第三版）上册》， 《工科数学分析（第三版）下册》， 高等教育出版社，张宗达主编 课程简介：

本课程的教学目的是使学生较系统的理解该课程的基本概念、基本理论、掌握基本方法，为后继课和进一步获取数学知 识奠定必要的数学基础。在传授知识的同时，着重培养学生抽象思维能力、逻辑推理能力、空间想象能力和自学能力，特别是综合运用所学知识去分析问题和解决问题能力。提高学生的素质，培育创新，创业精神。

第一学期讲授的主要内容：函数、极限、连续，一元函数微分学，一元函数积分，导数与定积分的应用；第二学期讲授的主要内容：多元函数微分学，多元函数积分学，无穷级数，常微分方程，复变函数初步、微分几何基础知识。 评分标准：作业——20%

期中考试——20% 期末考试——60%

Mathematics Analysis for Science and Technology Majors

Course Code: 08N1120211 08N1120212 Hours: 90 + 90 Credits: 5.5+5.5 Instructor: Hong Bai

Textbook: Zongda Zhang, Mathematical Analysis for Engineering I II, Higher Education Press Course Description:

Mathematics Analysis for Science and Technology Majors is one of the significant fundamental courses of nature science to every students majoring in engineering. The main content of this course is calculus. The required chapters are function, limits and continuity, derivative and differential, the mean theorems, indefinite integration, definite integration, the approach of derivative and definite integration, differential equation, Derivatives, Integration, the second type curve integral, the second-king surface integral, Vector and infinite series, initial of complex function. The fundamental knowledge of differential geometry is the significant content of the course.

Grading: Homework --------------20% Midterm exam-----------20% Final exam-----------------60% Syllabus: I. Function

1.1 Basic concepts

1.2 Elementary function 1.3 Examples

II.Limits and continuity 2.1 Limit of a sequence

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