Teaching Area

Equivalence relations and partitions. Functions, composition of functions, Invertible functions, one to one correspondence and cardinality of a set.

Permutations, cycle notation for permutations, even and odd permutations.

Definition and elementary properties of groups. Symmetries of a square, Dihedral groups, Quaternion groups (through matrices). Permutation group, alternating group, finite groups: , . The group of integers under addition modulo n and the group of units under multiplication modulo n.

Order of an element, order of a group, simple properties.

Orthogonal matrix and its properties. Rank of a matrix, inverse of a matrix, characterizations of invertible matrices. Row reduced and echelon forms, Normal form and congruence operations.

Solutions of systems of linear equations of the form and their applications

Sequences, bounded sequence, convergent sequence, limit of a sequence,

Limit theorems. Sandwich theorem. Nested interval theorem.

Monotone sequences, monotone convergence theorem.

Sub-sequences, divergence criteria. Monotone sub-sequence theorem (statement only).

Bolzano Weierstrass theorem for sequences.

Cauchy sequence, Cauchy’s convergence criterion, Cauchy’s 1st and 2nd limit theorems.

Lipschitz condition and Picard’s Theorem (Statement only).

General solution of homogeneous equation of second order, principle of superposition for homogeneous equation.

Wronskian: its properties and applications, linear homogeneous and non-homogeneous equations of higher order with constant coefficients.

Euler’s equation, method of undetermined coefficients.

Method of variation of parameters.