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Northeastern University

Linear Algebra

Northeastern University via XuetangX

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Overview

The course of linear algebra discusses the classic theory of linear relationships in algebra, which is an important fundamental course for science, engineering, economics and management majors for its basic concept, theory and methods have obvious abstraction, logicality and wide applicability. Since linear problems exist widely in various fields of science and technology, and some nonlinear problems can be transformed into linear problems under certain conditions, scientific computing problems are becoming increasingly important in engineering technology. With the development of information science, the course plays more and more important role in current times, and it has become a widely used mathematical tool in the field of natural science and engineering technology. The scope of linear algebra covers arrays, matrix, linear relation of vector groups, linear equations, matrix similar diagonalization, quadratic form, linear space and linear transformation.

This online course includes a video detailed explains the theory and application of linear algebra, proper exercises and supplemental materials. The course has the following characteristics:

1. It contains a complete set of teaching materials including more than 40 knowledge point videos, about 200 exercises, PPT, the explanation of typical examples, teaching calendar, study guide and so on.

2. It focuses on the systematization and coherence of knowledge, and aims to expanding students’ thinking. In the explanation of theory and examples, it focuses on inspiring and improving students’ learning by asking questions or pointing out ideas to arouse students’ initiative.

3.The teaching speech will be rigorous, precise and exact. Problem-solving focuses on standard procedures and clear ideas.

4. Pay attention to the correctness of thoughts and try to cultivate students’ international perspective.

This course will make the students to understand the basic theories and methods of linear algebra, cultivate students’ scientific computing ability, improve students’ abstract logical thinking and reasoning ability, and lay the essential foundation for expanding the knowledge of mathematics and learning related course theories. It can further improve students’ mathematics quality and develop students’ exploration spirit and practical innovation ability.


Syllabus

  • Chapter 1 Determinants
    • 1.1.1 The 2nd and 3rd Order Determinants
    • 1.1.2 The nth Order Determinants
    • 1.2.1 Cofactor Expansion Theorem and Transpose of Determinants
    • 1.2.2 Properties of Determinants
    • 1.3.1 Evaluations of Determinants 1
    • 1.3.2 Evaluations of Determinants 2
    • 1.4.1 Cramer's Rule
    • 1.4.2 Applications of Cramer's Rule
  • Chapter 2 Matrix
    • 2.1.1 Matrix Operations
    • 2.1.2 Matrix Multiplication
    • 2.2.1 The Transpose of a Matrix and Determinant of a Matrix
    • 2.2.2 The Inverse of Matrix and Its Properties
    • 2.3.1 The Calculation of Inverse Matrix
    • 2.3.2 Partitioned Matrices and Their Operations
    • 2.4.1 Elementary Operations and Elementary Matrices
    • 2.4.2 Application of Elementary Operations and Elementary Matrices
  • Chapter 3 Linear Dependence of Vector Set
    • 3.1.1 Vectors and its Operations
    • 3.1.2 Linear Combination
    • 3.2.1 Linear Dependence
    • 3.2.2 Determination of Linear Dependence
    • 3.3.1 Maximum Independent Subset
    • 3.3.2 Rank of Vector Set
    • 3.4.1 Rank of Matrix
    • 3.4.2 Method for Finding the Rank of Matrix and Vector Set
  • Mid-term Examination
    • Chapter 4 Linear Equations and Linear Spaces
      • 4.1.1 Decision of Solutions to Linear System
      • 4.1.2 Elimination Method for Solving Linear System
      • 4.1.3 Decision theorem of solution
      • 4.2.1 Structure of Solutions to Homogeneous System
      • 4.2.2 Structure of Solutions to Nonhomogeneous System
      • 4.3.1 Linear Space
      • 4.3.2 Subspaces
      • 4.4.1 Basis and Dimension
      • 4.4.2 Coordinate Vector and Isomorphic Vector Spaces
      • 4.4.3 Change of Basis and Change of Coordinate
      • 4.5 Linear Transformation
      • 4.6 Euclidean Space
    • Chapter 5 Eigenvalues and Eigenvectors
      • 5.1.1 Eigenvalues and Eigenvectors
      • 5.1.2 Properties of Eigenvalues and Eigenvectors
      • 5.2.1 Similar Matrices
      • 5.2.2 Diagonalization and its Conditions
      • 5.3 Real Symmtric Matrices and its Orthogonal Similar Diagonalization
    • Chapter 6 Quadratic Forms
      • 6.1.1 Quadratic Forms and Their Standard Forms
      • 6.1.2 Using Orthogonal Transformations to Deduce the Quadratic Forms to the Standard Forms
      • 6.2.1 Using the Method of Completing Squares to Reduce the Quadratic Forms to the Standard Forms
      • 6.2.2 Positive Definite Quadratic Forms
    • Final Examination

      Taught by

      Song Shuni, XueZhang, Jingyi Liu, and Shi Datao

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