GATE MA key terms

A set with a binary operation that is closed, associative, has an identity element, and gives every element an inverse. Groups model symmetry and are the central object of abstract algebra.

A set with two binary operations (addition and multiplication) where addition forms an abelian group and multiplication is associative and distributes over addition. Need not be commutative.

A commutative ring in which every nonzero element has a multiplicative inverse, so division (except by zero) is always possible. Examples include the rationals, reals, and complex numbers.

A set of vectors over a field, closed under vector addition and scalar multiplication, satisfying axioms such as distributivity and existence of a zero vector. The setting for linear algebra.

A linearly independent spanning set of a vector space. Every vector is a unique finite linear combination of basis elements, and all bases have the same cardinality, the dimension.

A map between vector spaces preserving addition and scalar multiplication. Once bases are chosen it is represented by a matrix, linking abstract maps to concrete computation.

The dimension of the image (column space) of a linear map or matrix. It equals the number of linearly independent rows and, via rank-nullity, complements the nullity.

A scalar λ for which some nonzero vector v satisfies Av = λv. Eigenvalues reveal invariant directions of a linear map and are roots of the characteristic polynomial.

Writing a matrix or linear operator as P D P⁻¹ where D is diagonal. Possible exactly when there is a basis of eigenvectors, e.g. for symmetric real matrices.

A pairing of two vectors returning a scalar, symmetric (or conjugate-symmetric), linear, and positive definite. It induces a norm and the notion of orthogonality.

The condition that two vectors have zero inner product. Orthogonal sets simplify projections, and orthonormal bases make coordinates easy to compute via inner products.

A scalar assigned to a square matrix that is zero exactly when the matrix is singular. It measures how the associated linear map scales volumes and appears in the characteristic polynomial.

A rectangular array of field elements encoding a linear map between finite-dimensional vector spaces with respect to chosen bases. Matrix multiplication corresponds to composition of maps.

A function for which small changes in input produce small changes in output; formally, preimages of open sets are open. Continuity is the basic regularity assumption in analysis.

The property that a function has a well-defined derivative, meaning it is locally well approximated by a linear map. Differentiability implies continuity but not conversely.

For a continuous function on [a,b] differentiable on (a,b), there exists c with f'(c) equal to the average slope. It links local derivatives to global change.

An expansion of a smooth function as an infinite sum of its derivatives at a point times powers of the displacement. It provides polynomial approximations with controlled remainder.

The integral defined as a limit of Riemann sums over partitions, existing for functions whose upper and lower sums agree. It generalises area under a curve.

Convergence of a sequence of functions where the rate is independent of the point. It preserves continuity and allows interchange of limits with integrals and derivatives.

Convergence of a sequence of functions at each individual point of the domain. Weaker than uniform convergence, it need not preserve continuity of the limit function.

A set equipped with a distance function satisfying positivity, symmetry, and the triangle inequality. It provides the setting for sequences, limits, and continuity in abstract spaces.

An integral over an unbounded interval or of an unbounded function, defined as a limit of proper integrals. Convergence must be checked separately from the integrand's behaviour.

A complex function that is complex differentiable in a neighbourhood of each point of its domain. Analytic functions are infinitely differentiable and equal to their power series.

The pair of partial differential equations relating the real and imaginary parts of a complex function that are equivalent to complex differentiability. They encode conformality of the map.

An integral of a complex function along a curve in the complex plane, computed by parametrising the path. It underlies Cauchy's theorem and residue calculus.

A result stating that the integral of a meromorphic function around a closed contour equals 2πi times the sum of residues of enclosed poles. It converts integrals into algebraic sums.

An equation involving an unknown function of one variable and its derivatives. Solutions are found by separation, integrating factors, or series and transform methods.

An equation relating an unknown function of several variables to its partial derivatives. Classical examples are the heat, wave, and Laplace equations, solved by separation of variables.

A differential equation together with conditions specifying the solution and possibly its derivatives at a single point. Existence and uniqueness are governed by Lipschitz-type conditions.

An integral transform mapping a function of time to a function of a complex parameter, converting linear differential equations into algebraic equations. Inversion uses complex contour integration.

A representation of a periodic function as a sum of sines and cosines. Coefficients are computed by integrating the function against these basis functions over one period.

A triple of a sample space, a sigma-algebra of events, and a probability measure assigning values in [0,1] with total mass one. The foundation for all probabilistic modelling.

A measurable function from a probability space to the real numbers. Its distribution describes the probabilities of the values it takes.

The probability-weighted average value of a random variable, computed as a sum or integral against its distribution. It is linear regardless of dependence between variables.

The expected squared deviation of a random variable from its mean, measuring spread. It equals the second moment minus the square of the first moment.

The probability of an event given that another event has occurred, defined as the joint probability divided by the conditioning event's probability. It underlies Bayes' theorem.

A pair of a vertex set and an edge set, possibly with weights or directions. Graphs model pairwise relations and are studied for connectivity, colouring, and traversal.

A connected acyclic graph. Any two vertices are joined by exactly one simple path, and a tree on n vertices has exactly n−1 edges.

A subgraph that is a tree and includes every vertex of the original connected graph. Minimum spanning trees minimise total edge weight and are found by greedy algorithms.

A closed walk traversing every edge of a graph exactly once. Such a circuit exists exactly when the graph is connected and every vertex has even degree.

A cycle passing through every vertex of a graph exactly once. Deciding whether one exists is NP-complete, unlike the edge-based Eulerian analogue.