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Course Coordinated by
IISc Bangalore
NPTEL
>>
Courses
>>
Mathematics
>> NOC:Linear Algebra (Video) >>
Introduction to Algebraic Structures  Rings and Fields
Question in Lecture
Week1
Introduction to Algebraic Structures  Rings and Fields
Definition of Vector Spaces
Examples of Vector Spaces
Definition of subspaces
Examples of subspaces
Week2
Examples of subspaces (continued)
Sum of subspaces
System of linear equations
Gauss elimination
Generating system , linear independence and bases
Week3
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
Week4
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)
Week5
Existence of a basis [continued]
Introduction to Linear Maps
Examples of Linear Maps
Linear Maps and Bases
Pigeonhole principle in Linear Algebra
Week6
Interpolation and the rank theorem
Examples
Direct sums of vector spaces
Projections
Direct sum decomposition of a vector space
Week7
Dimension equality and examples
Dual spaces
Dual spaces (continued)
Quotient spaces
Homomorphism theorem of vector spaces
Week8
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
Week9
Computational rules for matrices
Rank of a matrix
Computation of the rank of a matrix
Elementary matrices
Elementary operations on matrices
Week10
LR decomposition
Elementary Divisor Theorem
Permutation groups
Canonical cycle decomposition of permutations
Signature of a permutation
Week11
Introduction to multilinear maps
Multilinear maps (continued)
Introduction to determinants
Determinants (continued)
Computational rules for determinants
Week12
Properties of determinants and adjoint of a matrix
Adjointdeterminant theorem
The determinant of a linear operator
Determinants and Volumes
Determinants and Volumes (continued)
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