SYLLABUS- Solutions B.Sc. MATHEMATICS-IV, Vector Spaces & MatricesB.A./B.Sc. IV Semester"Paper-IVector spaces: Vector space, sub spaces, Linear combinations, linear spans, Sumsand direct sums.Bases and Dimensions: Linear dependence and independence, Bases anddimensions, Dimensions and subspaces, Coordinates and change of bases.Matrices: Idempotent, nilpotent, involutary, orthogonal and unitary matrices, singularand nonsingular matrices, negative integral powers of a nonsingular matrix; Trace of amatrix.Rank of a matrix: Rank of a matrix, linear dependence of rows and columns of amatrix, row rank, column rank, equivalence of row rank and column rank, elementarytransformations of a matrix and invariance of rank through elementary transformations,normal form of a matrix, elementary matrices, rank of the sum and product of twomatrices, 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-homogeneousequations, matrices of rotation and reflection.Real AnalysisB.A./B.Sc. IV Semester"Paper-IIContinuity and Differentiability of functions: Continuity of functions, Uniformcontinuity, Differentiability, Taylor's theorem with various forms of remainders.Integration: Riemann integral-definition and properties, integrability of continuousand monotonic functions, Fundamental theorem of integral calculus, Mean valuetheorems of integral calculus.Improper Integrals: Improper integrals and their convergence, Comparison test,Dritchlet's test, Absolute and uniform convergence, Weierstrass M-Test, Infinite integraldepending on a parameter.Sequence and Series: Sequences, theorems on limit of sequences, Cauchy'sconvergence criterion, infinite series, series of non-negative terms, Absoluteconvergence, 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 ofuniform convergence, Weierstrass M-Test, Abel's and Dritchlet's test, Convergence anduniform convergence of sequences and series of functions.Mathematical MethodsB.A./B.Sc. IV Semester"Paper-IIIIntegral Transforms: Definition, Kernel.Laplace Transforms: Definition, Existence theorem, Linearity property, Laplacetransforms of elementary functions, Heaviside Step and Dirac Delta Functions, FirstShifting Theorem, Second Shifting Theorem, Initial-Value Theorem, Final-ValueTheorem, 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 differentialand integro-differential equations using Laplace transforms. Dirichlet's condition.Fourier Transforms: Fourier Complex Transforms, Fourier sine and cosinetransforms, Properties of FourierTransforms, Inverse Fourier transforms.