Mathematics Optional - Algebra (Linear & Modern) Module

Vector spaces over ℝ and ℂ, linear dependence and independence;
Subspaces, bases, dimension; Linear transformations, rank and nullity;
Matrix of a linear transformation, algebra of matrices;
Row/column reduction, Echelon form, congruence and similarity;
Rank and inverse of a matrix; Solving linear systems;
Eigenvalues, eigenvectors, characteristic polynomial;
Cayley-Hamilton theorem, symmetric, Hermitian, orthogonal, and unitary matrices.

Groups, subgroups, cyclic groups, cosets, Lagrange's Theorem;
Normal subgroups, quotient groups, homomorphism of groups;
Basic isomorphism theorems, permutation groups, Cayley's theorem;
Rings, subrings, ideals, homomorphisms of rings;
Integral domains, principal ideal domains, Euclidean and UFDs;
Fields and quotient fields.

Linear programming problems;
Basic solution, basic feasible solution and optimal solution;
Graphical and simplex methods of solutions;
Duality;
Transportation problems;
Assignment problems.