Syllabus - GATE 2016
Mathematics (MA)
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, Skew-Hermitian and unitary
matrices; Finite dimensional inner product spaces, Gram-Schmidt
orthonormalization process, self-adjoint operators, definite forms.
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.
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.
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.
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.
Functional Analysis
Normed linear
spaces, Banach spaces, Hahn-Banach extension theorem, open mapping and closed
graph theorems, principle of uniform boundedness; Inner-product spaces, Hilbert
spaces, orthonormal bases, Riesz representation theorem, bounded linear
operators.
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.
Partial Differential
Equations
Linear and
quasilinear first order partial differential equations, 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.
Topology
Basic concepts of topology, bases,
subbases, subspace topology, order topology, product topology, connectedness,
compactness, countability and separation axioms, Urysohn’s Lemma.
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, , , distributions; Simple
linear regression.
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.
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