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Description of  Krishna's Fully Solved B.Sc. Mathematics-IV , Edition-2

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.