Math-IVB (Algebra)

Course Code: MM 254 | Credits: 4 | Prerequisite: First year Mathematics courses

Unit I:(Sets and Logic) Sets, relations, equivalence relations, partial order, mathematical induction and elements of mathematical logic.
Unit II:(Groups) Groups and examples, subgroups, the order of an element, cyclic groups, normal subgroups, permutation groups and quotient groups.
Unit III:(Homomorphisms) Group homomorphisms and isomorphisms, the fundamental theorem of group homomorphisms and its applications.
Unit IV:(Vector Spaces) Vector spaces and examples, subspaces, spanning sets, linear dependence and independence, basis, dimension and the sum of two subspaces.
Unit V:(Linear Transformations) Linear transformations and isomorphism; that every finite dimensional vector space is isomorphic to Rⁿ; the rank-nullity theorem, the dimension of quotient spaces, and the matrix of a linear transformation with examples.

Explain relations in full detail, revisit induction, and explain the elements of mathematical logic.
Explain the concept of a group with several examples, and through it the basic concepts of abstract mathematics.
Discuss which maps are useful in the study of groups and the concept of isomorphism, and explain proofs of basic theorems about groups.
Define a vector space and explain the concept of a basis via spanning sets and linearly independent sets.
Explain linear transformations between vector spaces, especially finite dimensional ones and the connection between matrices and linear maps, and solve related problems.

Gallian, Joseph A. Contemporary Abstract Algebra. Cengage, 2012
Herstein, I. N. Topics in Algebra. Wiley, 1975
Hoffman, Kenneth and Kunze, Ray. Linear Algebra. Pearson, 1971
Kumaresan, S. Linear Algebra: A Geometric Approach. Prentice Hall of India, 2004