ESSENTIALS OF ALGEBRA & TRIGONOMETRY

ESSENTIALS OF ALGEBRA & TRIGONOMETRY UGC MODEL BA/B SC (HONOURS) PART-I MATHEMATICS BMH 101 (a & b) ALGEBRA AND TRIGONOMETRY. ALGEBRA (Duration : Two Semesters/One Year) Mappings : Equivalence relations and partitions. Congruence modulo n. Symmetric. Skew symmetric. Hermitian and skew Hermitian matrices. Elementary operations on matrices. Inverse of a matrix. Linear independence of row and column matrices. Row rank, column rank and rank of a matrix. Equivalence of column and row ranks. Eigenvalues, eigenvectors and the characteristics equation of a matrix. Cayley Hamilton theorem and its use in finding inverse of a matrix. Applications of matrices to a system of linear (both homogeneous and non-homogeneous) equations. Theorems on consistency of a system of linear equations. Relations between the roots and coefficients of general polynomial equation in one variable. Transformation of equations. Descarte’s rule of signs. Solution of cubic equations (Cardon method). Biquadratic equations. Defintion of a group with examples and simple properties. Subgroups. Generation of groups. Cyclic groups. Coset decomposition. Lagrange’s theorem and its consequences. The fundamental theorem of homomorphism and Isomorphism. Normal subgroups. Quotient groups. The fundamental theorem of homomorphism. Permutation groups. Even and odd permutations. The alternating groups An. Cayley’s theorem. Introduction to rings, subrings, integral domains and fields. Characteristics of a ring. TRIGONOMETRY De Moivre’s theorem and its applications. Direct and inverse circular and hyperbolic functions, Logarithm of a complex quantity. Expansion of trigonometrical functions. Gregory’s series. Summation of series.