The main purpose of this course in the study of linear operators on finite dimensional vector spaces. The idea is to emphasize the simple geometric notions common to many parts of mathematics and its applications. Except for an occasional reference to undergraduate mathematics, the course will be self-contained. The algebraic co-ordinate free methods will be adopted through out the course. These methods are elegant and as elementary as the classical as coordinatized treatment. The scalar field will be arbitrary (even a finite field), however, in the treatment of vector spaces with inner products, special attention will be given to the real and complex cases. Determinants via the theory of multilinear forms. Variety of examples of the important concepts. The exercises will constitute significant asstion ; ranging from routine applications to ones which will extend the very best students.

INTENDED AUDIENCE : BSc / BE /ME/ MSc / PhD

PREREQUISITES : Language of Set Theory; Elementary Algebra and Calculus

Introduction to Algebraic Structures - Rings and Fields.

Definition of Vector Spaces.

Examples of Vector Spaces.

Definition of subspaces.

Examples of subspaces.

Examples of subspaces (continued).

Sum of subspaces.

System of linear equations.

Gauss elimination.

Generating system , linear independence and bases.

Examples of a basis of a vector space.

Review of univariate polynomials.

Examples of univariate polynomials and rational functions.

More examples of a basis of vector spaces.

Vector spaces with finite generating system.

Steinitzs exchange theorem and examples.

Examples of finite dimensional vector spaces.

Dimension formula and its examples.

Existence of a basis.

Existence of a basis (continued).

Existence of a basis (continued).

Introduction to Linear Maps.

Examples of Linear Maps.

Linear Maps and Bases.

Pigeonhole principle in Linear Algebra.

Interpolation and the rank theorem.

Examples.

Direct sums of vector spaces.

Projections.

Direct sum decomposition of a vector space.

Dimension equality and examples.

Dual spaces.

Dual spaces (continued).

Quotient spaces.

Homomorphism theorem of vector spaces.

Isomorphism theorem of vector spaces.

Matrix of a linear map.

Matrix of a linear map (continued).

Matrix of a linear map (continued).

Change of bases.

Computational rules for matrices.

Rank of a matrix.

Computation of the rank of a matrix.

Elementary matrices.

Elementary operations on matrices.

LR decomposition.

Elementary Divisor Theorem.

Permutation groups.

Canonical cycle decomposition of permutations.

Signature of a permutation.

Introduction to multilinear maps.

Multilinear maps (continued).

Introduction to determinants.

Determinants (continued).

Computational rules for determinants.

Properties of determinants and adjoint of a matrix.

Adjoint-determinant theorem.

The determinant of a linear operator.

Determinants and Volumes.

Determinants and Volumes (continued).

1. Artin, M. : Algebra, Prentice-Hall, 1994.

2. Halmos, P. R. : Finite-Dimensional Vector Spaces, Springer-Verlag, 1993.

3. Herstein, I. N. : Topics in Algebra, Wiley Eastern, 1987.

4. Hoffman, K. and Kunze, R. : Linear Algebra, Prentice-Hall, 1972.

5. Jacobson, N. : Basic Algebra, Vols. I & II, Hindustan Pub. Co., 1984.

6. Greub, W. : Linear Algebra, Springer-Verlag, GTM 97, (4-th edition) 1981.

Dilip P. Patil received B. Sc. and M. Sc. in Mathematics from the University of Pune in 1976 and 1978,
respectively. From 1979 till 1992 he studied Mathematics at School of Mathematics, Tata Institute of
Fundamental Research, Bombay and received Ph. D. through University of Bombay in 1989. Currently he is
a Professor of Mathematics at the Departments of Mathematics, Indian Institute of Science, Bangalore. At
present he is a Visiting Professor at the Department of Mathematics, IIT Bombay. He has been a Visiting
Professor at Ruhr-Universitt Bochum, Universitt Leipzig, Germany and several universities in Europe and
Canada. His research interests are mainly in Commutative Algebra and Algebraic Geometry.

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.

The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).

Date and Time of Exams: **23 October 2021** Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.

Registration url: Announcements will be made when the registration form is open for registrations.

The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.

Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.

Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IISc Bangalore. It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

## DOWNLOAD APP

## FOLLOW US