# GATE Mathematics Syllabus

## 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