Overview

COURSE OUTLINE: 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 co-ordinate 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. The exercises will constitute significant asstion ; ranging from routine applications to ones which will extend the very best students.

Syllabus

Lec01 Introduction to Algebraic Structures Rings and Fields.
Lec02 Defnition of Vector Spaces.
Lec03 Examples of Vector Spaces.
lec04 Defnition of subspaces.
Lec05 Examples of subspaces.
Lec06 Examples of subspaces continued.
Lec07 Sum of subspaces.
Lec08 System of linear equations.
lec09 Gauss elimination.
Lec10 Generating system , linear independence and bases.
Lec11 Examples of a basis of a vector space.
Lec12 Review of univariate polynomials.
Lec13 Examples of univariate polynomials and rational functions.
Lec14 More examples of a basis of vector spaces.
Lec15 Vector spaces with finite generating system.
Lec16 Steinitzs exchange theorem and examples.
Lec17 Examples of finite dimensional vector spaces.
Lec18 Dimension formula and its examples.
Lec19 Existence of a basis.
Lec20 Existence of a basis continued.
Lec21 Existence of a basis continued.
Lec22 Introduction to Linear Maps.
Lec23 Examples of Linear Maps.
Lec24 Linear Maps and Bases.
Lec25 Pigeonhole principle in Linear Algebra.
Lec26 Interpolation and the rank theorem.
Lec27 Examples.
Lec28 Direct sums of vector spaces.
Lec29 Projections.
Lec30 Direct sum decomposition of a vector space.
Lec31 Dimension equality and examples.
Lec32 Dual spaces.
Lec33 Dual spaces continued.
Lec34 Quotient spaces.
Lec35 Homomorphism theorem of vector spaces.
Lec36 Isomorphism theorem of vector spaces.
Lec37 Matrix of a linear map.
Lec38 Matrix of a linear map continued.
Lec39 Matrix of a linear map continued.
Lec40 Change of bases.
Lec41 Computational rules for matrices.
Lec42 Rank of a matrix.
Lec43 Computation of the rank of a matrix.
Lec44 Elementary matrices.
Lec45 Elementary operations on matrices.
Lec46 LR decomposition.
Lec47 Elementary Divisor Theorem.
Lec48 Permutation groups.
Lec49 Canonical cycle decomposition of permutations.
Lec50 Signature of a permutation.
Lec51 Introduction to multilinear maps.
Lec52 Multilinear maps continued.
Lec53 Introduction to determinants.
Lec54 Determinants continued.
Lec55 Computational rules for determinants.
Lec56 Properties of determinants and adjoint of a matrix.
Lec57 Adjoint determinant theorem.
Lec58 The determinant of a linear operator.
Lec59 Determinants and Volumes.
Lec60 Determinants and Volumes continued.