# InfoCoBuild

## Linear Algebra

Linear Algebra. Instructor: Prof. Dilip Patil, Department of Mathematics, IISc Bangalore. The main purpose of this course is 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 coordinate free methods will be adopted throughout 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. (from nptel.ac.in)

 Introduction to Algebraic Structures

 Lecture 01 - Introduction to Algebraic Structures - Rings and Fields Lecture 02 - Definition of Vector Spaces Lecture 03 - Examples of Vector Spaces Lecture 04 - Definition of Subspaces Lecture 05 - Examples of Subspaces Lecture 06 - Examples of Subspaces (cont.) Lecture 07 - Sum of Subspaces Lecture 08 - System of Linear Equations Lecture 09 - Gauss Elimination Lecture 10 - Generating System, Linear Independence and Basis Lecture 11 - Examples of a Basis of a Vector Space Lecture 12 - Review of Univariate Polynomials Lecture 13 - Examples of Univariate Polynomials and Rational Functions Lecture 14 - More Examples of a Basis of Vector Spaces Lecture 15 - Vector Spaces with Finite Generating System Lecture 16 - Steinitz Exchange Theorem and Examples Lecture 17 - Examples of Finite Dimensional Vector Spaces Lecture 18 - Dimension Formula and its Examples Lecture 19 - Existence of a Basis Lecture 20 - Existence of a Basis (cont.) Lecture 21 - Existence of a Basis (cont.) Lecture 22 - Introduction to Linear Maps Lecture 23 - Examples of Linear Maps Lecture 24 - Linear Maps and Bases Lecture 25 - Pigeonhole Principle in Linear Algebra Lecture 26 - Interpolation and the Rank Theorem Lecture 27 - Examples Lecture 28 - Direct Sums of Vector Spaces Lecture 29 - Projections Lecture 30 - Direct Sum Decomposition of a Vector Space Lecture 31 - Dimension Equality and Examples Lecture 32 - Dual Spaces Lecture 33 - Dual Spaces (cont.) Lecture 34 - Quotient Spaces Lecture 35 - Homomorphism Theorem of Vector Spaces Lecture 36 - Isomorphism Theorem of Vector Spaces Lecture 37 - Matrix of a Linear Map Lecture 38 - Matrix of a Linear Map (cont.) Lecture 39 - Matrix of a Linear Map (cont.) Lecture 40 - Change of Bases Lecture 41 - Computational Rules for Matrices Lecture 42 - Rank of a Matrix Lecture 43 - Computation of the Rank of a Matrix Lecture 44 - Elementary Matrices Lecture 45 - Elementary Operations on Matrices Lecture 46 - LR Decomposition Lecture 47 - Elementary Divisor Theorem Lecture 48 - Permutation Groups Lecture 49 - Canonical Cycle Decomposition of Permutations Lecture 50 - Signature of a Permutation Lecture 51 - Introduction to Multilinear Maps Lecture 52 - Multilinear Maps (cont.) Lecture 53 - Introduction to Determinants Lecture 54 - Determinants (cont.) Lecture 55 - Computational Rules for Determinants Lecture 56 - Properties of Determinants and Adjoint of a Matrix Lecture 57 - Adjoint-Determinant Theorem Lecture 58 - The Determinant of a Linear Operator Lecture 59 - Determinants and Volumes Lecture 60 - Determinants and Volumes (cont.)

 References Linear Algebra Instructor: Prof. Dilip Patil, Department of Mathematics, IISc Bangalore. The main purpose of this course is the study of linear operators on finite dimensional vector spaces.