### Description and Objective

Algebra-III is the study of vector spaces and linear transformations. The main objective of this course is to help students learn in a rigorous manner, the tools and methods essential for studying the solution spaces of problems in mathematics, engineering, the natural sciences, and social sciences and develop mathematical skills needed to apply these to the problems arising within their field of study; and to various real-world problems.

Prerequisite

Algebra-II

### Learning Outcomes

After the completion of the course, students will be able to identify and construct linear transformations of a matrix, characterize linear transformations as onto, one-to-one, can solve linear systems represented as linear transforms, can express linear transforms in other forms, such as matrix equations, and vector equations and characterize a set of vectors and linear systems using the concept of linear independence.

### Contents

1. Vector space, definition, examples, and properties
2. Subspaces
3. Linear combination and Linear Span of a set,
4. Linearly Dependent and Linearly Independent sets
5. Basis of a vector space
6. the dimension of a vector space
7. Intersections, sums and direct sums of subspaces
8. Quotient Spaces,
9. Change of basis, Linear transformation
10. Rank and Nullity of the linear transformation
11. Eigenvalues and eigenvectors
12. Dual spaces
13. Inner product Spaces with properties, Projection
14. Cauchy inequality, Orthogonal and orthonormal basis
15. Gram Schmidt process and diagonalization.

### Recommended Books

1. Dar K. H, Linear Algebra, (1st edition, 2007).

2. Kolman B., Hill D. R., Introductory Linear Algebra, (Pearson/Prentice Hall 8th edition, 2005)

### Suggested Books

1. Cherney D., Denton T., Thomas R., and Waldron A. Linear Algebra, (Davis California, 1st edition 2013)

2. Anton H., Rorres C., Elementary Linear Algebra: Applications Version, (John Wiley & Sons, 11th edition 2014.)

3. Grossman S. I. Elementary Linear Algebra, (Cengage Learning, 5th edition, 2004.)

RESEARCH PROJECT /PRACTICALS/LABS/ASSIGNMENTS

The projects assigned in this course follow as assignments the exercises related to the topics from the suggested book.

### Assessment Criteria

Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)

Mid-Term Exam:   30

Final-Term Exam: 50

### Key Dates and Time of Class Meeting

Wednesday                                              02:00 PM-03:30 PM

Thursday                                                  03:30 PM-05:00 PM

Commencement of Classes                                                   January 13, 2020

Mid Term Examination                                                            March 09-13, 2020

Final Term Examination                                                          May 04-08, 2020

Declaration of Result                                                              May 19, 2020