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GATE Mathematics Syllabus

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Subject Code: MA

Course Structure

Sections/Units Topics
Section A Linear Algebra
Section B Complex Analysis
Section C Real Analysis
Section D Ordinary Differential Equations
Section E Algebra
Section F Functional Analysis
Section G Numerical Analysis
Section H Partial Differential Equations
Section I Topology
Section J Probability and Statistics
Section K Linear programming

Course Syllabus

Section A: 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
    • Unitary matrices
  • Finite dimensional inner product spaces −
    • Gram-Schmidt orthonormalization process
    • Self-adjoint operators, definite forms

Section B: 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 C: 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
    • Gauss
  • Metric spaces −
    • Compactness
    • Completeness
    • Weierstrass approximation theorem
  • Lebesgue measure −
    • Measurable functions
  • Lebesgue integral −
    • Fatou’s lemma
    • Dominated convergence theorem

Section D: 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 E: 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, and field extensions

Section F: 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

Section G: 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

Section H: 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

Section I: Topology

  • Basic concepts of topology
  • Bases
  • Subbases
  • Subspace topology
  • Order topology
  • Product topology
  • Connectedness
  • Compactness
  • Countability
  • Separation axioms
  • Urysohn’s lemma

Section J: 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

Section H: 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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