Week 1
Introduction to Algebraic Structures - Rings and Fields.
Definition of Vector Spaces.
Examples of Vector Spaces.
Definition of subspaces.
Examples of subspaces.
Week 2
Examples of subspaces (continued).
Sum of subspaces.
System of linear equations.
Gauss elimination.
Generating system , linear independence and bases.
Week 3
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.
Week 4
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).
Week 5
Existence of a basis (continued).
Introduction to Linear Maps.
Examples of Linear Maps.
Linear Maps and Bases.
Pigeonhole principle in Linear Algebra.
Week 6
Interpolation and the rank theorem.
Examples.
Direct sums of vector spaces.
Projections.
Direct sum decomposition of a vector space.
Week 7
Dimension equality and examples.
Dual spaces.
Dual spaces (continued).
Quotient spaces.
Homomorphism theorem of vector spaces.
Week 8
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.
Week 9
Computational rules for matrices.
Rank of a matrix.
Computation of the rank of a matrix.
Elementary matrices.
Elementary operations on matrices.
Week 10
LR decomposition.
Elementary Divisor Theorem.
Permutation groups.
Canonical cycle decomposition of permutations.
Signature of a permutation.
Week 11
Introduction to multilinear maps.
Multilinear maps (continued).
Introduction to determinants.
Determinants (continued).
Computational rules for determinants.
Week 12
Properties of determinants and adjoint of a matrix.
Adjoint-determinant theorem.
The determinant of a linear operator.
Determinants and Volumes.
Determinants and Volumes (continued).
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