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September 23, 2023

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Description

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)

Course Curriculum

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

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