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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)

Lecture 39 - Matrix of a Linear Map (cont.)

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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.)