Mathematics-IV

SYLLABUS- Solutions B.Sc. MATHEMATICS-IV, Vector Spaces & Matrices
B.A./B.Sc. IV Semester–Paper-I
Vector spaces: Vector space, sub spaces, Linear combinations, linear spans, Sums
and direct sums.
Bases and Dimensions: Linear dependence and independence, Bases and
dimensions, Dimensions and subspaces, Coordinates and change of bases.
Matrices: Idempotent, nilpotent, involutary, orthogonal and unitary matrices, singular
and nonsingular matrices, negative integral powers of a nonsingular matrix; Trace of a
matrix.
Rank of a matrix: Rank of a matrix, linear dependence of rows and columns of a
matrix, row rank, column rank, equivalence of row rank and column rank, elementary
transformations of a matrix and invariance of rank through elementary transformations,
normal form of a matrix, elementary matrices, rank of the sum and product of two
matrices, inverse of a non-singular matrix through elementary row transformations;
equivalence of matrices.
Applications of Matrices: Solutions of a system of linear homogeneous equations,
condition of consistency and nature of the general solution of a system of linear non-homogeneous
equations, matrices of rotation and reflection.
Real Analysis
B.A./B.Sc. IV Semester–Paper-II
Continuity and Differentiability of functions: Continuity of functions, Uniform
continuity, Differentiability, Taylor's theorem with various forms of remainders.
Integration: Riemann integral-definition and properties, integrability of continuous
and monotonic functions, Fundamental theorem of integral calculus, Mean value
theorems of integral calculus.
Improper Integrals: Improper integrals and their convergence, Comparison test,
Dritchlet’s test, Absolute and uniform convergence, Weierstrass M-Test, Infinite integral
depending on a parameter.
Sequence and Series: Sequences, theorems on limit of sequences, Cauchy’s
convergence criterion, infinite series, series of non-negative terms, Absolute
convergence, tests for convergence, comparison test, Cauchy’s root Test, ratio Test,
Rabbe’s, Logarithmic test, De Morgan’s Test, Alternating series, Leibnitz’s theorem.
Uniform Convergence: Point wise convergence, Uniform convergence, Test of
uniform convergence, Weierstrass M-Test, Abel’s and Dritchlet’s test, Convergence and
uniform convergence of sequences and series of functions.
Mathematical Methods
B.A./B.Sc. IV Semester–Paper-III
Integral Transforms: Definition, Kernel.
Laplace Transforms: Definition, Existence theorem, Linearity property, Laplace
transforms of elementary functions, Heaviside Step and Dirac Delta Functions, First
Shifting Theorem, Second Shifting Theorem, Initial-Value Theorem, Final-Value
Theorem, The Laplace Transform of derivatives, integrals and Periodic functions.
Inverse Laplace Transforms: Inverse Laplace transforms of simple functions,
Inverse Laplace transforms using partial fractions, Convolution, Solutions of differential
and integro-differential equations using Laplace transforms. Dirichlet’s condition.
Fourier Transforms: Fourier Complex Transforms, Fourier sine and cosine
transforms, Properties of FourierTransforms, Inverse Fourier transforms.