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Linear Algebra

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Overview

To present a careful treatment of the principal topics of linear algebra and to illustrate the power of the subject through a variety of applications.  Topics including: systems of linear equations, determinants, matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, and quadratic forms. Linear algebra is suitable for all students majoring in science or mathematics, or any other courses of study requiring it. 


After completing this course, students will be able to understand the basic methods and techniques in linear algebra. This course also helps students develop their abilities in abstract thinking and logical reasoning. 


Linear Algebra serves as a prerequisite for many courses, including numerical analysis, ordinary differential equations, partial differential equations, regression analysis, financial mathematics and financial engineering and etc.


Syllabus

  • Chapter 1 Gaussian Elimination and Matrices
    • 01 Gaussian Elimination
    • 02 Linear Equations
    • 03 Vectors in R^n
    • 04 Matrix Multiplication
    • 05 Elementary Matrices
    • 06 The Interplay
    • 07 LU Factorization
    • 08 Inverses
    • 09 Inverses Properties
    • 10 Example Transposes
    • 11 Partitioned Matrices
  • Chapter 2 Vector Spaces
    • 12 Vector Spaces and Examples
    • 13 Subspaces
    • 14 Linear Independence
    • 15 Basis
    • 16 Maximal Linearly Independent Subset
    • 17 Solving Ax=0
    • 18 Solving Ax=b
    • 19 Four Fundamental Subspaces
  • Chapter 3 Orthogonality
    • 20 Inner Product
    • 21 Orthogonal Vectors
    • 22 Projection Onto Lines
    • 23 Projection Onto Column Space
    • 24 Least-Squares
    • 25 Orthonormal Basis
    • 26 Gram-Schmidt Orthogonalization Procedure
    • 27 QR Factorization
  • Chapter 4 Determinants
    • 28 Determinants Definition
    • 29 Determinants Properties
    • 30 Determinants Formulas
    • 31 Determinants Examples
    • 32 Inverses
    • 33 Volume
    • 34 Cramer's Rule
  • Chapter 5 Eigenvalues and Eigenvectors
    • 35 Introduction
    • 36 Eigenvalues and Eigenvectors
    • 37 Diagonalization of a Matrix
    • 38 Diagonalization of a Matrix- Application
    • 39 System of Differential Equations
  • Chapter 6 Positive Definite Matrices and Quadratic Forms
    • 40 Complex Matrices
    • 41 Linear Transformations
    • 42 Similarity Transformations
    • 43 Jordan Form
    • 44 Quadratic Forms
    • 45 Positive Definite Matrices
    • 46 Positive Semi-Definite Matrices
    • 47 Congruence Transformation
    • 48 Singular Value Decomposition
    • 49 Application
    • 50 Minimum Principles
  • Final Exam

    Taught by

    Southern University of Science and Technology

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