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