Introduction to Abstract and Linear Algebra: Sourav Mukhopadhyay

LECTURE	TOPIC	DESCRIPTION
1.	Set Theory	Definition of Set, Representation of set, empty set, universal set, subset.
2.	Set Operations	Union, Intersection, properties of union and intersection, complementation.
3.	Set Operations (cont...)	De Morgan's law, set difference, symmetric difference, Venn diagram.
4.	Set of sets	Power set, cardinality of the power set, partition of a set, cartesian product, relation.
5.	Binary relation	Definition of binary relation, examples, relfextive, symmetric, transitive and equivalence relation.
6.	Equivalence relation	Equivalence classes, partition, Zn.
7.	Mapping	Definition of mapping, domain, co-domain, image, range, one-to-one, onto, bijective function, composition of mapping.
8.	Permutation	Bijective mapping, cycle, equipotent set, equivalence relation, denumerable (enumerable), countable set.
9.	Binary Composition	Binary operator, closure, commutative, associative, example of binary composition.
10.	Groupoid	Algebric structure, Groupoid, commutative groupoid, identity and inverse element, semigroup, monoid, quasigroup, group.
11.	Group	Group, finite groups, abelian group, uniqueness of identity and inverse.
12.	Order of an element	Integral power of an element, order of an element, properties of order.

13.	Subgroup	Induced binary composition, subgroups, unique identity, example of subgroups, necessary and sufficient conditions for subgroups.
14.	Cyclic Group	Sufficient condition for subgroup, centre of a group, centraliser of an element, cyclic subgroup generated by an element, cyclic group, generator.
15.	Subgroup Operations	Intersection and union of two subgroups, product of two subgroups, definition of left cosets.
16.	Left Cosets	Left cosets of a subgroup, disjoint left cosets, left cosets as equivalence classes, any two left cosets are having same cardinality, lagrange theorem on order of a subgroup.
17.	Right Cosets	Definition, equivalence classes of right cosets, number of distinct left cosets and number of distinct right cosets are same, index of a subgroup.
18.	Normal Subgroup	Definition of normal subgroups, necessary and sufficient condition for normal subgroups, Quotient group, homomorphism, isomorphism.
19.	Rings	Definition of rings, commutative rings, rings with identity, examples of rings, polynomial rings.
20.	Field	Divisors of zeros, Integral Domain, Field.
21.	Vector Spaces	External composition, definition of vector spaces, vector spaces over real field
22.	Sub-Spaces	Properties of vector spaces, vector sub-spaces, necessary and sufficient condition for sub- spaces, linear sum of sub-spaces.
23.	Linear Span	Linear combination of vectors, linear span, linear dependency and independency.
24.	Basis of a Vector Space	Finite dimensional vector space, definition of basis, Replacement theorem, any two bases have same number of vectors.

25.	Dimension (or Rank) of a Vector space	Definition of rank of a vector space, dimension of null space is zero, extension theorem, co- ordinate of a vector with respect to a ordered basis.
26.	Complement of subspace	Dimension of subspaces, sum of two subspaces, direct sum of two subspaces, dimension of sum of two subspaces.
27.	Linear Transformation	Definition of linear mapping, examples, properties of linear mapping, kernel of a linear mapping is a sub-space, one-to-one linear mapping.
28.	Linear Transformation (cont...)	Image of a linear mapping, subspace, dimension of kernel and image of a linear mapping, nullity and rank theorem of a linear mapping.
29.	More on linear mapping	Composition of linear mapping, inverse of linear mapping, isomorphism, isomorphic, two same dimensional (finite) vector spaces (U and V say) are isomorphic (U ~ V).
30.	Linear Space	V ~ F^n, addition of two linear mapping, scalar multiplication of two linear mapping, linear space of linear mappings, matrix representation of linear mapping.
31.	Rank of a matrix	definition of rank of a matrix, maximum order of nonzero minor, elementary row and column operations, row reduced echelon form, row equivalence
32.	Rank of a matrix (cont....)	Elementary matrices, row and columns operations is same as multiplying with elementary matrices, rank of a matrix is same as rank of the row reduced or columns reduced matrix, normal form.
33.	System of linear equations	augmented matrix, rank of augmented matrix, consistency, inconsistencies, no solution, unique solution and infinitely many solution
34.	Row rank and Column rank	Row space and Column space of a matrix, row rank, column rank
35.	Eigen value of a matrix	Characteristic equation, Cayley-Hamilton theorem, eigen value of a matrix, product of eigen values

36.	Eigen Vector	Eigen value of a non singular matrix, eigen vectors, eigen vector corresponding to unique eigen value, independence of eigen vectors
37.	Geometric multiplicity	Algebric and geometric multiplicity of eigen value, simple eigen value, regular eigen value
38.	More on eigen value	Eigen values of a real symmetric matrix are real, eigen values of a orthogonal martix
39.	Similar matrices	Eigen value of real orthogonal matrix, similar relation between matrices, two similar matrix have same eigen values
40.	Diagonalisable	Definition of diagonalisable, conditions of diagonalisable, examples

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