GATE XE key terms

A set of vectors is linearly independent when no nontrivial linear combination of them equals the zero vector. Equivalently, each vector lies outside the span of the others, so the set carries no redundant direction.

The dimension of the column space, equal to the number of linearly independent rows or columns. It bounds how many independent equations a system supplies and determines whether a square matrix is invertible.

A system Ax = b is consistent when at least one x satisfies it. This holds exactly when b lies in the column space of A, that is, when the rank of A equals the rank of the augmented matrix.

The set of all vectors x satisfying Ax = 0. It is a subspace whose dimension, the nullity, equals the number of columns minus the rank, and it measures the non-uniqueness of solutions.

A scalar λ for which Ax = λx has a nonzero solution x. Found from the characteristic equation det(A − λI) = 0; real symmetric matrices always have real eigenvalues and orthogonal eigenvectors.

A nonzero vector whose direction is unchanged by multiplication by a matrix, so Ax = λx for the corresponding eigenvalue λ. Eigenvectors belonging to distinct eigenvalues of a symmetric matrix are mutually orthogonal.

Every square matrix satisfies its own characteristic polynomial: substituting the matrix into p(λ) = det(A − λI) gives the zero matrix. Useful for computing inverse powers and reducing matrix polynomials.

A matrix is diagonalizable when it has enough linearly independent eigenvectors to form a basis, giving A = PDP⁻¹ with D diagonal. Distinct eigenvalues guarantee this; repeated eigenvalues require checking eigenvector counts.

The derivative of a multivariable function with respect to one variable while holding the others fixed. It measures how the function changes along a coordinate direction and underpins the gradient and total differential.

The vector of partial derivatives of a scalar field. It points in the direction of steepest increase, is perpendicular to level curves or surfaces, and vanishes at interior extrema of differentiable functions.

The matrix of first partial derivatives of a vector-valued function, mapping input perturbations to output changes. Its determinant appears in change-of-variables formulas for multiple integrals.

A scalar field formed by the dot product of the del operator with a vector field. It measures local source or sink strength; a field with zero divergence everywhere is called solenoidal.

A vector field formed by the cross product of the del operator with a vector field. It measures local rotation or circulation density; a field with zero curl everywhere is called irrotational.

Relates the circulation of a vector field around a closed curve to the flux of its curl through any surface bounded by that curve. It generalizes the fundamental theorem of calculus to surfaces.

States that the flux of a vector field through a closed surface equals the volume integral of its divergence over the enclosed region. It converts surface integrals into volume integrals.

An integral transform mapping a time function to a function of a complex parameter s, converting linear differential equations into algebraic equations. Initial conditions enter directly, and solutions are recovered by inverse transform.

An operation combining two functions by integrating one against a reversed, shifted copy of the other. In Laplace and Fourier analysis it becomes ordinary multiplication, simplifying responses to arbitrary inputs.

The ratio of output to input Laplace transforms for a linear time-invariant system with zero initial conditions. Its poles and zeros determine stability, damping and frequency response.

A representation of a periodic function as a sum of sines and cosines at integer multiples of the fundamental frequency. Coefficients are obtained from orthogonality integrals over one period.

An integral transform decomposing a non-periodic function into a continuous spectrum of frequencies. It converts differentiation into multiplication by jω and convolution into multiplication.

States that the total energy of a signal equals the energy summed over its Fourier coefficients or integrated over its spectral density. It expresses conservation of energy between time and frequency domains.

Two logarithmic graphs of a system's frequency response: magnitude in decibels and phase in degrees, both against log frequency. Straight-line asymptotes reveal poles, zeros, gain margin and phase margin.

An algebraic test on the coefficients of a characteristic polynomial that counts roots in the right half of the complex plane, using a Routh array. It determines closed-loop stability without factoring the polynomial.

The factor by which loop gain can be increased before a feedback system becomes marginally stable, read where phase crosses minus 180 degrees. Usually expressed in decibels.

The additional phase lag, measured at the gain crossover frequency, that would drive a feedback system to marginal stability. It indicates relative stability and damping of the closed-loop response.

An iterative root-finding scheme using the tangent line: xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ). It converges quadratically near a simple root but can diverge if the derivative is small.

A numerical integration formula approximating area by summing trapezoids between sample points. Its error is proportional to the square of the step size for a single application.

A numerical integration formula fitting parabolas through pairs of subintervals, giving higher accuracy than the trapezoidal rule. It requires an even number of intervals and has fourth-order error.

An expansion of a smooth function about a point using its derivatives there. Truncating after the first-order term underlies linearization and the error terms of numerical methods.

A second-order tensor whose components give the traction on any plane through a point. Its symmetry follows from moment equilibrium, and its principal values are the principal stresses.

The normal stress acting on a plane free of shear stress, found as an eigenvalue of the stress tensor. Maximum and minimum principal stresses bound the stress state at a point.

A graphical construction plotting normal stress against shear stress for all plane orientations through a point. It gives principal stresses, maximum shear stress and orientation angles directly.

A symmetric tensor describing deformation, with diagonal terms for normal strain and off-diagonal terms for shear. Small-strain theory relates it to displacement gradients.

A linear relation between stress and strain through elastic constants such as Young's modulus and shear modulus. Valid within the elastic limit, it generalizes to a fourth-order stiffness tensor for anisotropic materials.

The negative ratio of transverse strain to axial strain under uniaxial loading. Typical metals lie near 0.3; the value approaches 0.5 for incompressible materials and is negative for auxetics.

The stress at which a material begins to deform plastically, often reported as the 0.2 percent offset stress for metals without a distinct yield point. It marks the end of elastic behaviour.

The ratio of a material's failure strength to the working stress expected in service. Values above one provide margin against uncertainty in loads, material properties and analysis.