Algebra

Unit 1 · Matrix Algebra

Elementary operations of matrices, inverse of a matrix; special types of matrices — transpose, symmetric and skew-symmetric, Hermitian and skew-Hermitian; echelon and normal form, rank of a matrix; application of matrices to a system of linear (homogeneous and non-homogeneous) equations, theorems on consistency; eigenvalues and eigenvectors and the relation between them, process of finding them, Cayley-Hamilton theorem and its use in finding the inverse.

Unit 2 · Set Theory & Functions

Sets, subsets, set operations and laws of set theory, Venn diagrams, finite and infinite sets, counting principle, empty and power sets, classes of sets, set identities, generalized union and intersection. Relations and functions — product set, composition and types of relations, partitions, equivalence relations (congruence modulo), partial-ordering relations, functions and their types, inverse and composite functions, modular arithmetic and basic properties of congruences.

Unit 3 · Theory of Equations

Symmetric functions of the roots of an equation, root of a multiplicity, synthetic division, greatest common divisors, relation between roots and coefficients of general polynomial equations in one variable, transformation of equations, Descartes' rule of signs, solutions of cubic equations (Cardan method), biquadrate equation.

Unit 4 · Group Theory

Definition and properties of a group, abelian groups, examples of groups, subgroups and examples, cosets and their properties, Lagrange's theorem and its applications, normal subgroups and their properties, simple groups, factor groups.