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

Week 1

- Introduction to Algebraic Structures - Rings and Fields.
- Definition of Vector Spaces.
- Examples of Vector Spaces.
- Definition of subspaces.
- Examples of subspaces.

• Week 2

- Examples of subspaces (continued).
- Sum of subspaces.
- System of linear equations.
- Gauss elimination.
- Generating system , linear independence and bases.

• Week 3

- 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.

• Week 4

- 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).

• Week 5

- Existence of a basis (continued).
- Introduction to Linear Maps.
- Examples of Linear Maps.
- Linear Maps and Bases.
- Pigeonhole principle in Linear Algebra.

• Week 6

- Interpolation and the rank theorem.
- Examples.
- Direct sums of vector spaces.
- Projections.
- Direct sum decomposition of a vector space.

• Week 7

- Dimension equality and examples.
- Dual spaces.
- Dual spaces (continued).
- Quotient spaces.
- Homomorphism theorem of vector spaces.

• Week 8

- 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.

• Week 9

- Computational rules for matrices.
- Rank of a matrix.
- Computation of the rank of a matrix.
- Elementary matrices.
- Elementary operations on matrices.

• Week 10

- LR decomposition.
- Elementary Divisor Theorem.
- Permutation groups.
- Canonical cycle decomposition of permutations.
- Signature of a permutation.

• Week 11

- Introduction to multilinear maps.
- Multilinear maps (continued).
- Introduction to determinants.
- Determinants (continued).
- Computational rules for determinants.

• Week 12

- Properties of determinants and adjoint of a matrix.
- Adjoint-determinant theorem.
- The determinant of a linear operator.
- Determinants and Volumes.
- Determinants and Volumes (continued).

**LIST OF TEXTS / REFERENCE**

- Artin, M. : Algebra, Prentice-Hall, 1994.
- Halmos, P. R. : Finite-Dimensional Vector Spaces, Springer-Verlag, 1993.
- Herstein, I. N. : Topics in Algebra, Wiley Eastern, 1987.
- Hoffman, K. and Kunze, R. : Linear Algebra, Prentice-Hall, 1972.
- Jacobson, N. : Basic Algebra, Vols. I & II, Hindustan Pub. Co., 1984.
- Greub, W. : Linear Algebra, Springer-Verlag, GTM 97, (4-th edition) 1981.

- The exam is optional for a fee.
**Exams will be on 23 April 2017** **Time: Shift 1:**9am-12 noon;**Shift 2:**2pm-5pm- Any one shift can be chosen to write the exam for a course.
**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.

**CERTIFICATE:**

- Final score will be calculated as : 25% assignment score + 75% final exam score
- 25% assignment score is calculated as 25% of average of best 8 out of 12 assignments
- E-Certificate will be given to those who register and write the exam and score greater than or equal to 40% final 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