S11_MTS203_Linear_algebra

=MTS203: Linear Algebra=

Program:
BS(CS)

Semester:
Spring 2011

Instructor:
A. Raza

**Course Lead:**
Dr. N. Touheed

**Credit Hours:**
3 credit hours.

Prerequisite(s):
MTS101 (Calculus-1)

Course Description:
This introductory course on linear algebra will provide an opportunity to the students to become familiar with the fundamental patterns and structures of mathematical reasoning as used in mathematics, science and engineering. Teaching methods include classroom lectures, multimedia sessions, and student interaction over technical problems.

Course Objectives:
Successful students will be able to: work and how abstract axiomatic systems are developed and applied, and how their consequences are analyzed in form of mathematical theorems. a variety of viewpoints in the light of introductory matrix algebra and theory of linear operators on vector spaces. For the purposes of applications of linear algebras of square matrices and linear operators this course ensures that the students understand algorithms for Diagonalization and Gram-Schmidt Orthogonalization Procedure, Computing Powers of Matrices, and Least Square Methods as they are used in science and engineering.
 * Understand how proper mathematical definitions are stated and used in mathematical
 * Study a system of linear equations and associated allied problems from

Books:
__**Textbook:**__ __**Reference Books:**__
 * Elementary Linear Algebra with Application Version 8th Ed, Howard Anton & Chris Rorres, Publishers:John-Wiley c1994.
 * 1) Linear Algebra 2nd Ed Serge Lang, Publishers:Addison-Wesley c1978
 * 2) Linear Algebra: Schaum’s Outline Series, S.Lipschutz, Publishers: McGraw-Hill c1981
 * 3) Linear Algebra 3rd Printing, Kenneth Hoffman & Ray Kunze, Publishers:Prentice-Hall c1964

Grading Policy:
The grading policy followed for this course is given underneath: Assignment Course File Viva Term-1 Term-2 Terminal Exam ||= 5% 5% 5% 5% 20% 20% 40% ||
 * __**Assessment**__ ||= __**Weight (% of total score)**__ ||
 * Quizzes

Course Topics (Salient outlines):
Following is a rough dissemination of the topics into an approximate number of lectures. a) Consistency (Existence & Non-Existence), Uniqueness & Non-Uniqueness of Solutions; b) A Quick Review of Elementary Matrix Operations and Properties of Determinants; c) Elementary Row Operations and Echelon and Row Reduced Echelon Systems and Matrices; d) Geometrical Interpretation of Solutions: Points, Lines, Planes, and Hyperplanes; d) Gaussian Elimination and Backward Substitution Algorithm and (if possible) Gauss-Seidel Iteration Algorithm || 5 || a) A Brief Introduction of the Concept of Fields: Real & Complex; b) Motivation and Definition of Abstract Vector Spaces: Presentation of Concrete Examples of Vector Spaces; c) Linear Independence & Dependence; d) Linear Span, Basis and Dimension of Vector Spaces: Concrete Examples and Calculations; e) Subspaces: Unions, Intersections, Sums and Direct Sums; f) Vector Space Interpretation of Solutions of Linear Systems of Equations: Affine Space || 6 || a) Definitions and Examples of Linear Transformations; b) Kernel (Null Space) & Image of a Linear Transformation; c) Rank and Nullity of a Linear Transformation: Rank- Nullity Theorem & Concrete Calculations of Rank and Nullity; d) Change of Basis and Transition Matrix; e) Matrix Representation of a Linear Transformation; f) Linear Operator; g) Similarity Transformation; h) Homomorphism & Isomorphism of Vector Spaces; i) Definition of Linear Algebras of Real Square Matrices and Linear Operators on Vector Spaces: Concrete Examples. || 6 || a) Characteristic Polynomial of a Linear Operator: Cayley-Hamilton Theorem; b) Definitions of Eigenvalues and Eigenvectors: The Eigenvalue Problems and their concrete examples and calculations; c) Diagonalization and Diagonalizabilty Problem: Computing Powers of Matrices and (if possible) Least Square Methods. || 4 || a) Motivation and Definitions of Norm & Normed Spaces: Concrete Examples; b) Motivation and Definitions of Real & Complex Inner Product Spaces: Concrete Examples; c) Angles between vector in Real Vector Spaces; d) Orthogonal Vectors & Orthogonal Complements of Subspaces; Ortho-normal Systems of Basis Vectors and their Computational Advantages; e) Real Symmetric Matrices and Orthogonalizability Problem: Gram-Schmidt Orthogonalization Algorithm and Orthogonal Diagonalization || 7 ||
 * __**Topics**__ || __**Approx. # of lectures**__ ||
 * __**Systems of Linear Homogeneous and Non-Homogeneous Equations:**__
 * __**Vector Spaces: Real and Complex:**__
 * __**Linear Transformations:**__
 * __**Eigenvalue Problems for Linear Operators:**__
 * **__Normed & Inner Product Spaces:__**