《Linear Algebra》课程教学大纲

《Linear Algebra》课程教学大纲（三号黑体）

一、课程基本信息（四号黑体）

二、课程目标（四号黑体）"Linear Algebra" is a mathematical course studying vector spaces and linear transformations, which are widely used in applied science and technology. The topics in the course will cover the following aspects: linear equations, vector equations and matrix equations, linear transformations, matrix operations, determinants, vector spaces, eigenvalues and eigenvectors, orthogonality, symmetric matrices and quadratic forms, etc. After learning the course, the students will develop the abilities to analyze and solve the practical problems related with linear algebra and vectors.

（一）总体目标：（小四号黑体）

（以三维目标即知识与技能、过程与方法、情感态度与价值观的形式反映核心素养观念和内容，其中核心素养不仅关注学生“当下发展”，更关注学生“未来发展”所需要的正确价值观念、必备品格和关键能力，即把知识、技能和过程、方法提炼为能力，把情感态度、价值观提炼为品格）（五号宋体）

（二）课程目标：（小四号黑体）

（课程目标规定某一阶段的学生通过课程学习以后，在发展德、智、体、美、劳等方面期望实现的程度，它是确定课程内容、教学目标和教学方法的基础。）（五号宋体）

课程目标1：Master matrix, determinant, space vector and other basic theories and basic knowledge through systematic learning

1. Master the calculation of determinant, the elementary transformation of matrix, the definition and calculation of matrix rank, using the elementary transformation of matrix.

2. Solving equations and inverse matrices, linear dependence of vector groups, eigenvalues and eigenvectors, etc.

课程目标2：Through systematic study, master determinant calculation, matrix operation, elementary transformation, and apply to the solution of linear equations, quadratic form and linear transformation, can solve some mathematical problems.

 1．Ability to apply course content to research and analyze complex science and engineering problems.

    2. To understand the general application of linear algebra in the energy industry, and closely related to the development of advanced materials science, enhance learning ability and learning awareness, enhance social responsibility.

（要求参照《普通高等学校本科专业类教学质量国家标准》，对应各类专业认证标准，注意对毕业要求支撑程度强弱的描述，与“课程目标对毕业要求的支撑关系表一致）（五号宋体）

（三）课程目标与毕业要求、课程内容的对应关系（小四号黑体）

表1：课程目标与课程内容、毕业要求的对应关系表 （五号宋体）

（大类基础课程、专业教学课程及开放选修课程按照本科教学手册中各专业拟定的毕业要求填写“对应毕业要求”栏。通识教育课程含通识选修课程、新生研讨课程及公共基础课程，面向专业为工科、师范、医学等有专业认证标准的专业，按照专业认证通用标准填写“对应毕业要求”栏；面向其他尚未有专业认证标准的专业，按照本科教学手册中各专业拟定的毕业要求填写“对应毕业要求”栏。）

三、教学内容（四号黑体）

（具体描述各章节教学目标、教学内容等。实验课程可按实验模块描述）

第一章 Linear Equations（小四号黑体）

1.教学目标 （五号宋体）

 Learning knowledges about linear equations.

2.教学重难点

Problems:

1. Think of the relations among linear equations and vector and matrix equations?

2. Think of the difference between linear dependence and linear independence?

3. What is row reduced echelon form?

4. What is a linear transformation?

5. Think of the solution sets of a general linear systems?

3.教学内容

Section 1: Introduction and Linear equations

Class 1: Introduction, Basic concepts of systems of linear equations

Class 2: Row reduction and echelon forms I

Class 3: Row reduction and echelon forms II, exercises

Section 2: Vector equations and matrix equations

Class 4: Definition of vector equations, exercises

Class 5: Definition of matrix equations, exercises

Class 6: Solution sets of linear systems

Section 3: Linear Independence and Linear Transformations

Class 7: Definition of linear dependence and independence

Class 8: Introduction to linear transformations

Class 9: The matrix of a linear transformation

4.教学方法

 Method of lecture and questions between classes and tutoring after class.

5.教学评价

Through the teaching of this chapter, students can master linear equations.

第二章 Matrix Algebra （小四号黑体）

1.教学目标 （五号宋体）

Learning knowledges about vector equations and matrix equations.

2.教学重难点

Problems:

1. Think of the properties about matrix operations?

2. Think of the characterizations of invertible matrix?

3. What is a singular matrix?

4. What is the algorithm for finding the inverse of a matrix?

3.教学内容

Section 1: Introduction to Matrix Operations

Class 1: Definition of matrix operations, addition, multiplication, transpose

Class 2: Inverse of a matrix, exercises

Class 3: Characterizations of invertible matrix

Section 2: More on matrix

Class 4: Partitioned matrix, exercises

Class 5: Matrix equations, exercises

Class 6: Elementary matrix, symmetric matrix, exercises

4.教学方法

Method of lecture and questions between classes and tutoring after class.

5.教学评价

Through the teaching of this chapter, students can master vector equations and matrix equations.

第三章 Determinants （小四号黑体）

1.教学目标 （五号宋体）

 Learning knowledges about determinants, matrix operations and linear transformations.

2.教学重难点

Problems:

1. What is a determinant, how many ways to calculate it?

2. Think of the relations between matrix and determinant?

3. What is a adjoint of a matrix?

4. What is the use of the Cramer’s rule?

5. What are general properties for determinant calculation?

3.教学内容

Section 1: Introduction to Determinants

Class 1: Definition of determinants, the expansion theorem

Class 2: Properties of determinants, exercises

Class 3: Cramer’s rule, exercises

Class 4: More calculation on determinants

4.教学方法

Method of lecture and questions between classes.

5.教学评价

Through the teaching of this chapter, students can masterdeterminants, matrix operations and linear transformations.

第四章 Vector Spaces（小四号黑体）

1.教学目标 （五号宋体）

 Learning knowledges about vector spaces.

2.教学重难点

Problems:

1. What is a vector space and subspace?

2. Think of the difference between null space and column space?

3. What is the use of rank?

4. What is the dimension of a vector space?

5. What is the procedure for a change of basis and coordinate transformation?

3.教学内容

Section 1: Introduction to Vector Spaces

Class 1: Definition of vector spaces and subspaces, properties

Class 2: Definition of null space, column space, exercises

Class 3: Linearly independent sets, bases

Section 2: Coordinate transformation

Class 4: Definition of coordinate system, coordinate transformation

Class 5: Dimension of a vector space, the basis theorem

Class 6: Rank, the rank theorem, exercises

Class 7: Change of basis, exercises

4.教学方法

Method of lecture and questions between classes and exercises after classes.

5.教学评价

Through the teaching of this chapter, students can master vector spaces.

第五章 Eigenvalues and Eigenvectors（小四号黑体）

1.教学目标 （五号宋体）

 Learning knowledges about eigenvalues and eigenvectors.

2.教学重难点

Problems:

1. What are eigenvalues and eigenvectors?

2. What is the procedure to solve eigen-problems?

3. What is the condition for a matrix which can be diagonalized or not?

4. What are the properties for the characteristic equation?

3.教学内容

Section 1: Eigen-problems

Class 1: Definition of eigenvalues and eigenvectors

Class 2: The characteristic equation, exercises

Class 3: Diagonalization of a matrix I

Class 4: Diagonalization of a matrix II, exercises

Class 5: More on eigen-problems and diagonalization, exercises

4.教学方法

Method of lecture and questions between classes and exercises after classes.

5.教学评价

Through the teaching of this chapter, students can master Learning knowledges about eigenvalues and eigenvectors.

第六章 Orthogonality（小四号黑体）

1.教学目标 （五号宋体）

 Learning knowledges about orthogonality.

2.教学重难点

Problems:

1. What is inner product?

2. What is orthogonality?

3. What is the Schmidt process?

4. What is the use of orthogonal projection?

5. Think of the difference between vector space and inner product space?

3.教学内容

Section 1: Inner Product and Orthogonality

Class 1: Definition of inner product, length, orthogonality

Class 2: Orthogonal sets, orthogonal projection, exercises

Class 3: The Gram-Schmidt process, exercises

Class 4: Inner product spaces

4.教学方法

Method of lecture and questions between classes and exercises after classes.

5.教学评价

Through the teaching of this chapter, students can master orthogonality.

第七章 Symmetric Matrices and Quadratic Forms（小四号黑体）

1.教学目标 （五号宋体）

 Learning knowledges about symmetric matrices and quadratic forms, etc.

2.教学重难点

Problems:

1. Think of the difference in the diagonalization between common matrix and symmetric matrix?

2. What is the procedure for the diagonalization of a symmetric matrix?

3. What are the different types of quadratic forms?

4. What is the procedure for solving a quadratic form?

5. What is the use of the principal axes theorem for quadratic forms?

3.教学内容

Section 1: Symmetric matrices and quadratic forms

Class 1: Diagonalization of symmetric matrix, exercises

Class 2: The spectral theorem, exercises

Class 3: Definition of quadratic form

Class 4: The principal axes theorem, exercises

4.教学方法

Method of lecture and questions between classes and exercises after classes.

5.教学评价

Through the teaching of this chapter, students can master symmetric matrices and quadratic forms, etc.

四、学时分配（四号黑体）

表2：各章节的具体内容和学时分配表（五号宋体）

五、教学进度（四号黑体）

表3：教学进度表（五号宋体）

六、教材及参考书目（四号黑体）

（电子学术资源、纸质学术资源等，按规范方式列举）（五号宋体）

 Textbooks：D. C. Lay，《Linear Algebra and Its Applications》，(Publishing House of Electronics Industry)，2010.

References：

1. Linear Algebra, by Z.M. Tang, et al., (Science Press, 2011). (In Chinese)

2. Linear Algebra (6ed), by Tongji University, (Higher Education Press, 2014). (In Chinese)

3. Linear Algebra with Applications (9ed), by S.J. Leon, (China Machine Press, 2017).

4. Introduction to Linear Algebra (5ed), by L.W. Johnson, et al., (China Machine Press, 2012).

5. Linear Algebra done right (2ed), by S. Axler, (Beijing World Publishing Corporation, 2008).

七、教学方法 （四号黑体）

（讲授法、讨论法、案例教学法等，按规范方式列举，并进行简要说明）（五号宋体）

1.  Theoretical class teaching,

2.  practice and discussion class,

3.   Q&A and problem communication,

4.  electronic resources and mathematics experiment learning guidance.

八、考核方式及评定方法（四号黑体）

（一）课程考核与课程目标的对应关系 （小四号黑体）

表4：课程考核与课程目标的对应关系表（五号宋体）

（二）评定方法 （小四号黑体）

1．评定方法 （五号宋体）

（例：平时成绩：10%，期中考试：30%，期末考试60%，按课程考核实际情况描述）（五号宋体）

Course Attendance and Performance: 10%

Homework and Quiz: 20%

Midterm Exam: 30%

Final Exam: 40%

2．课程目标的考核占比与达成度分析 （五号宋体）

表5：课程目标的考核占比与达成度分析表（五号宋体）

（三）评分标准 （小四号黑体）