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**Course Code**: B11003Y-A01

**Course Name**: Linear Algebra ǁ (A)

**Credits**: 4.0

**Level**: Undergraduate

**Lecture Time**: 80 hrs

**Pre-requisite**: Linear Algebra I

**Instructors**:

**Course Description**

This course is a required basic course for all undergraduates in all majors. It is required that students master the basic knowledge and methods of linear algebra through learning this course. This course is divided into three types: type A (Mathematical), type B (Science and Technology) and type C (Engineering).

These three types have the same basic content (see the teaching content and class hours). However, the specific calculation and applications are paid more attention in type C; there is more theoretical analysis in type B, while the foundation for type A contains rigorous theoretical proof.

Type A is suitable for those students who will continue to study mathematics, physics, computer and other subjects. Type B is suitable for those students who will study any majors other than mathematics in the future. Type C is suitable for those students who will study any major other than mathematics, physics, and computer in the future.

Through this course, students should meet the following basic requirements: (1) Master the basic knowledge of linear algebra; (2) Understand and form the thinking of linear algebra to lay foundation for further study on basic and applied mathematics courses; (3) have the basic ability to solve problems from different backgrounds by using the knowledge and methods of linear algebra.

**Topics and Schedule **

1. Space and form (10 hrs)

1.1. Abstract vector space, dimension and base

1.2. Dual space

1.3. Bilinear and quadratic forms

2. Linear operator (12 hrs)

2.1. Linear mapping and operator algebra of vector space

2.2. Invariant subspace and feature vector

2.3. Jordan standard form

3. Vector space with scalar product (14 hrs)

3.1. Euclidean vector space

3.2. Hermite vector space

3.3. Linear operators on the space with scalar product

3.4. Complexification and realification

3.5. Orthogonal polynomial

4. Affine space and Euclidean space (10 hrs)

4.1. Affine space, Euclidean space

4.2. Group and geometry

4.3. Finite metric space with exponential

5. Quadric surface (11 hrs)

5.1. Quadratic function

5.2. Quadric surface in affine space and Euclidean space

5.3. Projective space and quadric surface in it

6. Tensor (8 hrs)

6.1. Preliminary calculation of tensor

6.2. Convolution, symmetrization and alternizer of tensor

6.3. Exterior algebra

**Teaching methods**

1. Two times a week and 90 minutes each time for teaching, 50 minutes for classroom teaching, 10 minutes for rest and 40 minutes for classroom teaching.

2. One time a week and 100 minutes each time for exercises and discussion.

**Grading**

The assessment will include the class routine performance, the mid-term examination and the final examination, which will count for 30%, 30% and 40% each. The class routine performance includes homework, classroom discipline, attendance, discussion and so on. The mid-term and the final examination are close-book examinations.

**Textbook**

Kostrikin, Introduction to Algebra, the second volume

**References**

[1] Shirov, Introduction to the Linear Space (second edition), Higher Education Press

[2] D. C. Lay, Linear Algebra and its Applications, Machinery Industry Press

[3] Apostol. T. M , Linear Algebra and its Applications, People's Posts and Telecommunications Press

[4] Axler. S, We Should Learn Linear Algebra Like This, People's Posts and Telecommunications Press