# 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

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