GATE MA flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Rank–nullity theorem

For a linear map T from an n-dimensional space, dim of kernel plus dim of image equals n. Equivalently rank(A) + nullity(A) equals the number of columns of A.

Cayley–Hamilton theorem

Every square matrix satisfies its own characteristic polynomial. This lets you express high powers of A as linear combinations of lower powers, useful for computing inverses and matrix functions.

Sylvester's criterion

A real symmetric matrix is positive definite exactly when all its leading principal minors are positive. Equivalently, all its eigenvalues are positive and it defines a strict local minimum of a quadratic form.

Singular value decomposition

Any real matrix A factors as UΣVᵀ with U and V orthogonal and Σ diagonal of nonnegative singular values, the square roots of eigenvalues of AAᵀ. Rank equals the count of nonzero singular values.

Cauchy–Riemann equations

For f = u + iv, analyticity requires uₓ = v_y and u_y = −vₓ. When the partial derivatives are continuous, these equations are also sufficient for differentiability at a point.

Residue at a simple pole

If f has a simple pole at z₀, the residue equals the limit of (z − z₀)f(z) as z tends to z₀. The residue theorem then gives the contour integral as 2πi times the sum of enclosed residues.

Liouville's theorem

Every bounded entire function is constant. A standard consequence is the fundamental theorem of algebra: any nonconstant complex polynomial has at least one root in the complex plane.

Cauchy's integral formula

For f analytic on and inside a simple closed contour C, the value at an interior point z₀ is (1/2πi) times the contour integral of f(z)/(z − z₀) around C.

Hahn–Banach theorem

A bounded linear functional defined on a subspace of a normed space extends to the whole space without increasing its norm. It guarantees enough functionals to separate points and supports duality theory.

Riesz representation theorem

On a Hilbert space, every bounded linear functional has the form f(x) = inner product of x with a unique vector y, and the operator norm of f equals the norm of y.

Open mapping theorem

A bounded linear operator from one Banach space onto another is an open map. A corollary is the bounded inverse theorem: a bounded bijection between Banach spaces has a bounded inverse.

Uniform boundedness principle

Also called Banach–Steinhaus: a family of bounded linear operators on a Banach space that is pointwise bounded is uniformly bounded in operator norm. It yields results about divergence of Fourier series.

Picard–Lindelöf theorem

The initial value problem y′ = f(x, y), y(x₀) = y₀ has a unique solution near x₀ provided f is continuous and satisfies a Lipschitz condition in y there.

Wronskian test

For solutions of a linear homogeneous ODE, the Wronskian determinant is either identically zero or never zero on an interval. Nonzero Wronskian means the solutions are linearly independent.

Variation of parameters

A method for a particular solution of a linear nonhomogeneous ODE. Using two independent solutions y₁, y₂ and the Wronskian W, it builds the particular integral from integrals of forcing divided by W.

Stability of linear systems

For x′ = Ax, the zero solution is asymptotically stable exactly when every eigenvalue of A has strictly negative real part. Purely imaginary eigenvalues give neutral, bounded oscillations instead.

Method of characteristics

For a first-order quasilinear PDE, solutions are constant along curves satisfying dx/a = dy/b = du/c. Integrating these characteristic equations reduces the PDE to an ODE along each curve.

Classification of second-order PDEs

For Au_xx + Bu_xy + Cu_yy + lower terms = 0, the discriminant B² − 4AC decides type: negative means elliptic, zero means parabolic, positive means hyperbolic. Type can vary with position.

d'Alembert solution

The wave equation u_tt = c²u_xx on the whole line has general solution u = f(x − ct) + g(x + ct). Initial displacement and velocity determine f and g by half-sum and half-integral formulas.

Maximum principle

A nonconstant harmonic function on a bounded domain attains its maximum and minimum only on the boundary. This immediately gives uniqueness of the Dirichlet problem for Laplace's equation.

Newton–Raphson convergence

The iteration x − f(x)/f′(x) converges quadratically near a simple root, roughly doubling correct digits per step. It can fail or diverge if the derivative is zero or the start is poor.

Simpson's rule error

Composite Simpson's 1/3 rule has error −(b − a)h⁴f⁗(ξ)/180, so it is exact for cubics. It requires an even number of subintervals and smooth fourth derivative.

Gauss–Legendre quadrature

An n-point Gauss–Legendre rule chooses optimal nodes and weights so that it integrates exactly every polynomial of degree up to 2n − 1, giving higher accuracy than equally spaced Newton–Cotes rules.

Lagrange interpolation error

Interpolating f at n + 1 distinct nodes by a polynomial of degree at most n gives error f⁽ⁿ⁺¹⁾(ξ)/(n + 1)! times the product of (x − xᵢ), for some ξ in the interval.

Bayes' theorem

The conditional probability of A given B equals P(B|A)P(A) divided by P(B). It updates a prior probability into a posterior using the likelihood of the observed event.

Variance shortcut formula

Variance equals the second moment minus the square of the mean: Var(X) = E[X²] − (E[X])². For aX + b, variance scales by a² and is unaffected by the shift b.

Central limit theorem

For independent identically distributed variables with finite mean and variance, the standardised sample sum converges in distribution to the standard normal, justifying normal approximations for large samples.

Unbiased sample variance

The estimator dividing the sum of squared deviations by n − 1 has expectation σ². Dividing by n instead gives a biased estimator whose expectation is σ²(n − 1)/n.

Strong duality in LP

If a linear program has a finite optimal value, so does its dual, and the two optimal objective values coincide. This underlies the simplex stopping rule and sensitivity analysis.

Complementary slackness

At optimality, a positive primal variable forces its dual constraint to hold with equality, and a positive dual variable forces its primal constraint tight. It certifies optimality of a pair of solutions.

Balanced transportation problem

A transportation problem is balanced when total supply equals total demand. A basic feasible solution then has exactly m + n − 1 basic cells, where m and n are the numbers of sources and destinations.

Mean value theorem

If f is continuous on a closed interval and differentiable inside it, some interior point has derivative equal to the average slope across the interval. Rolle's theorem is the equal-endpoint case.