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MTH501 : Linear Algebra

Course Overview

Course Synopsis

This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices. The goals of this subject are, how we can use Linear Algebra and its numerical applications in different fields.

Course Learning Outcomes

Upon completing this course students should be able :

  • To master the techniques for solving systems of linear equations.
  • To introduce matrix algebra as a generalization of single-variable algebra of high school.
  • To build on the background in Euclidean space and formalize it with vector space theory.
  • To relate linear methods to other areas of mathematics such as calculus and differential equations.
  • To develop an appreciation for how linear methods are used in a variety of applications.

Course Calendar

1 Introduction
2 Introduction to Matrices.
3 System Of Linear Equations.
4 Row reduction and Echelon Form of a Matrix.
5 Vector Equations.
6 Matrix Equations.
7 Solution Set of Linear Equations.
8 Linearly Dependent and Linearly Independent Sets.
9 Linear Transformations.
10 The Matrix of Linear Transformations.
11 Matrix Algebra.
12 Inverse of a Matrix.
13 Characterisation of Invertible Matrices.
14 Partitioning of a Matrix.
15 Matrix Factorization.
16 Iterative Solution of Linear Systems.
17 Introduction to Determinants.
18 Properties of Determinants.
19 Cramer's rule, Volume and Linear Transformations.
20 Vector Spaces.
21 Null Spaces,Column Spaces and Linear Transformations.
22 Bases for a vector Space.
23 Coordinate systems.
24 Dimension of a Vector Space.
25 The Rank Theorem and Invertible Matrix Theorem.
26 Change of Bases of a Vector Space.
27 Applications of vector spaces to Difference Equations.
28 Eigenvalues of a Matrix.
29 Characteristic Equation of a Matrix.
30 Diagonalization of a Matrix.
31 Eigen Vectors of linear Transformation.
32 Complex Eigenvalues and Vectors.
33 Discrete Dynamical Systems.
34 Applications to Differential Equations.
35 Iterative Estimates for Eigenvalues.
36 Revision
37 Revision ( system of Linear Equations).
38 Inner Product and orthogonal vectors.
39 Orthogonal Sets.
40 Orthogonal Projections.
41 Gram-Schmidt Process.
42 Least Squares Problems.
43 Inner Product Spaces.
44 Applications of Inner Product Spaces.
45 Revision of the Course and Vector Spaces.
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