# GATE Mathematics MA Syllabus 2019 WWW.RAHBARS.COM

### Section 1: Linear Algebra

Finite dimensional vector spaces

Linear transformations and their matrix representations, rank

Systems of linear equations, eigenvalues and eigenvectors, minimal polynomial, Cayley-Hamilton Theorem, diagonalization, Jordan-canonical form, Hermitian, SkewHermitian and unitary matrices

Finite dimensional inner product spaces, Gram-Schmidt orthonormalization process, self-adjoint operators, definite forms

## GATE Mathematics MA Syllabus 2019

### Section 2: Complex Analysis

Analytic functions, conformal mappings, bilinear transformations

Complex integration: Cauchy’s integral theorem and formula

Liouville’s theorem, maximum modulus principle

Zeros and singularities

Taylor and Laurent’s series

Residue theorem and applications for evaluating real integrals

### Section 3: Real Analysis

Sequences and series of functions, uniform convergence, power series, Fourier series, functions of several variables, maxima, minima

Riemann integration, multiple integrals, line, surface and volume integrals, theorems of Green, Stokes and Gauss

Metric spaces, compactness, completeness, Weierstrass approximation theorem

Lebesgue measure, measurable functions

Lebesgue integral, Fatou’s lemma, dominated convergence theorem

## GATE Mathematics MA Syllabus 2019

### Section 4: Ordinary Differential Equations

First order ordinary differential equations, existence and uniqueness theorems for initial value problems, systems of linear first order ordinary differential equations, linear ordinary differential equations of higher order with constant coefficients

Linear second-order ordinary differential equations with variable coefficients

Method of Laplace transforms for solving ordinary differential equations, series solutions (power series, Frobenius method);

Legendre and Bessel functions and their orthogonal properties

### Section 5: Algebra

Groups, subgroups, normal subgroups, quotient groups, and homomorphism theorems, automorphisms

Cyclic groups and permutation groups, Sylow’s theorems and their applications

Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domains, Principle ideal domains, Euclidean domains, polynomial rings and irreducibility criteria

Fields, finite fields, field extensions

### Section 6: Functional Analysis

Normed linear spaces, Banach spaces, Hahn-Banach extension theorem, open mapping, and closed graph theorems, the principle of uniform boundedness

Inner-product spaces, Hilbert spaces, orthonormal bases, Riesz representation theorem, bounded linear operators

## GATE Mathematics MA Syllabus 2019

### Section 7: Numerical Analysis

Numerical solution of algebraic and transcendental equations: bisection, secant method, Newton-Raphson method, fixed point iteration

Interpolation: error of polynomial interpolation, Lagrange, Newton interpolations

Numerical differentiation

numerical integration: Trapezoidal and Simpson rules

numerical solution of systems of linear equations: direct methods (Gauss elimination, LU decomposition)

iterative methods (Jacobi and Gauss-Seidel)

Numerical solution of ordinary differential equations

initial value problems: Euler’s method, Runge-Kutta methods of order 2

## GATE Mathematics MA Syllabus 2019

### Section 8: Partial Differential Equations

Linear and quasilinear first order partial differential equations, the method of characteristics

Second order linear equations in two variables and their classification

Cauchy, Dirichlet and Neumann problems

solutions of Laplace, wave in two-dimensional Cartesian coordinates, Interior and exterior Dirichlet problems in polar coordinates

Separation of variables method for solving wave and diffusion equations in one space variable

Fourier series and Fourier transform and Laplace transform methods of solutions for the above equations

## GATE Mathematics MA Syllabus 2019

### Section 9: Topology:

Basic concepts of topology, bases, subbases, subspace topology, order topology, product topology, connectedness, compactness, countability and separation axioms, Urysohn’s Lemma

### Section 10: Probability and Statistics

Probability space, conditional probability, Bayes theorem, independence, Random variables, joint and conditional distributions, standard probability distributions and their properties (Discrete uniform, Binomial, Poisson, Geometric, Negative Binomial, Normal, Exponential, Gamma, Continuous uniform, Bivariate normal, Multinomial), expectation, conditional expectation, moments

Weak and strong law of large numbers, central limit theorem

Sampling distributions, UMVU estimators, maximum likelihood estimators; Interval estimation

Testing of hypotheses, standard parametric tests based on normal,x2, t, F distributions

Simple linear regression

## GATE Mathematics MA Syllabus 2019

### Section 11: Linear programming

Linear programming problem and its formulation, convex sets and their properties, graphical method, basic feasible solution, simplex method, big-M and two-phase methods

Infeasible and unbounded LPP’s, alternate optima

Dual problem and duality theorems, dual simplex method and its application in post-optimality analysis

Balanced and unbalanced transportation problems, Vogel’s approximation method for solving transportation problems

Hungarian method for solving assignment problems

### Some Important Link for GATE 2019

GATE Previous Exam (Sample Paper)

GATE 2019 Code Wise Syllabus

Official Website

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