Engineering Mathematics
Engineering Mathematics covers the linear algebra, calculus and probability that GATE CS asks about. Every worked example is checked by a small program: exact fractions for matrices, Simpson's rule for integrals, and exact enumeration for probabilities, so the numbers in the lessons and answers can be trusted.
1Linear Algebra
- Matrices, Determinants and RankMatrix operations and special matrices, determinants and their properties, the inverse, rank by row reduction, and exact computations with fractions.
- Systems of Linear EquationsWriting a system as Ax = b, Gaussian elimination, when a system has a unique solution, infinitely many or none (rank test), homogeneous systems, and LU decomposition.
- Eigenvalues and EigenvectorsThe eigenvalue equation, the characteristic polynomial, finding eigenvectors, properties (trace, determinant, powers, inverses, triangular and symmetric matrices), the Cayley–Hamilton theorem and diagonalization.
2Calculus
- Limits, Continuity and DifferentiationLimits and standard limits, L'Hôpital's rule, continuity and differentiability, Rolle's and the mean value theorem, and finding maxima and minima.
- IntegrationStandard integrals, substitution and integration by parts, properties of definite integrals (symmetry and the a − x trick), areas between curves, and checking answers with Simpson's rule.
3Probability and Statistics
- ProbabilitySample spaces and events, the axioms, counting-based probability, conditional probability and independence, the law of total probability, and Bayes' theorem.
- Random Variables and DistributionsDiscrete and continuous random variables, expectation and variance, and the distributions in the GATE syllabus: Bernoulli, binomial, Poisson, geometric, uniform, exponential and normal.
- Descriptive StatisticsMean, median and mode, range, variance and standard deviation (population and sample), how shifting and scaling data changes them, and correlation and the regression line.