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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.