Linear Algebra

Por: Swayam . en: , ,

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