# Virtual University of Pakistan

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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

 Topic Lecture Resource Page Introduction 1 Elementary Linear Algebra and its Application 1-5 Introduction to Matrices. 2 (3rd edition) 6-17 System Of Linear Equations. 3 By David C. Lay 18-29 Row reduction and Echelon Form of a Matrix. 4 30-43 Vector Equations. 5 44-55 Matrix Equations. 6 56-65 Solution Set of Linear Equations. 7 66-73 Linearly Dependent and Linearly Independent Sets. 8 81-88 Quiz No. 1 Linear Transformations. 9 89-225 The Matrix of Linear Transformations. 10 98-107 Matrix Algebra. 11 121-133 Inverse of a Matrix. 12 134-143 Characterization of Invertible Matrices. 13 144-149 Partitioning of a Matrix. 14 150-157 Matrix Factorization. 15 158-167 Iterative Solution of Linear Systems. 16 168-201 Assignment # 1 Introduction to Determinants. 17 202-207 Properties of Determinants. 18 208-216 Cramer"s rule, Volume and Linear Transformations. 19 217-222 Vector Spaces. 20 232-241 Null Spaces, Column Spaces and Linear Transformations. 21 242-252 Bases for a vector Space. 22 253-261 Midterm Examination Coordinate systems. 23 262-271 Dimension of a Vector Space. 24 272-278 The Rank Theorem and Invertible Matrix Theorem. 25 278-286 Change of Bases of a Vector Space. 26 287-289 Applications of vector spaces to Difference Equations. 27 290-301 Eigenvalues of a Matrix. 28 317-325 Quiz No. 2 Characteristic Equation of a Matrix. 29 326-334 Diagonalization of a Matrix. 30 335-343 Eigen Vectors of linear Transformation. 31 343-350 Assignment # 2 Complex Eigenvalues and Vectors. 32 351-358 Discrete Dynamical Systems. 33 358-364 Applications to Differential Equations. 34 369-373 Iterative Estimates for Eigenvalues. 35 379-384 GDB Problem Revision 36 385-386 Revision (system of Linear Equations). 37 386-387 Inner Product and orthogonal vectors. 38 387-399 Orthogonal Sets. 39 400-402 Orthogonal Projections. 40 402-407 Null Spaces, Column Spaces and Linear Transformations. 41 413-417 Gram-Schmidt Process for finding orthogonal Bases of a Vector Space. 42 418-423 Inner Product Spaces. 43 443-450 Applications of Inner Product Spaces. 44 452-454 Revision of the Course and Vector Spaces. 45 455-458 Final Term Exams Note: Any kind of change in the course calandar can be done during semester. So please keep update yourself from 'Overview' on VULMS.