Lecture 49 of 60Week 10
Canonical cycle decomposition of permutations
Prof. Dilip P. Patil · IISc Bangalore
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- Linear Algebra, IISc Bangalore (at NPTEL)
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All the lectures
- Introduction to Algebraic Structures - Rings and Fields
- Definition of Vector Spaces
- Examples of Vector Spaces
- Definition of subspaces
- Examples of subspaces
- Examples of subspaces (continued)
- Sum of subspaces
- System of linear equations
- Gauss elimination
- Generating system , linear independence and bases
- 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
- 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)
- Existence of a basis [continued]
- Introduction to Linear Maps
- Examples of Linear Maps
- Linear Maps and Bases
- Pigeonhole principle in Linear Algebra
- Interpolation and the rank theorem
- Examples
- Direct sums of vector spaces
- Projections
- Direct sum decomposition of a vector space
- Dimension equality and examples
- Dual spaces
- Dual spaces (continued)
- Quotient spaces
- Homomorphism theorem of vector spaces
- 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
- Computational rules for matrices
- Rank of a matrix
- Computation of the rank of a matrix
- Elementary matrices
- Elementary operations on matrices
- LR decomposition
- Elementary Divisor Theorem
- Permutation groups
- Canonical cycle decomposition of permutations
- Signature of a permutation
- Introduction to multilinear maps
- Multilinear maps (continued)
- Introduction to determinants
- Determinants (continued)
- Computational rules for determinants
- Properties of determinants and adjoint of a matrix
- Adjoint-determinant theorem
- The determinant of a linear operator
- Determinants and Volumes
- Determinants and Volumes (continued)