Glossary
Short, original definitions for higher-math vocabulary. Jump via the list or scroll. Terms are linked from topics and explainers across the site.
Basis
A linearly independent set of vectors that spans a vector space; every vector has unique coordinates relative to a basis.
Bayes’ theorem
A rule relating P(H|E) to P(E|H), the prior P(H), and P(E). Used to update probabilities after observing evidence.
Conditional probability
P(A|B) is the probability of A given that B occurred; probability reweighted to the subspace B.
Continuity
A function is continuous at a if the limit as x→a equals f(a). Informally, the graph does not jump at a.
Convergence (sequence/series)
A sequence converges if terms approach a limit. A series converges if its sequence of partial sums converges.
Definite integral
The signed accumulation of f from a to b, written ∫_a^b f(x) dx; equal to net signed area when f is integrable.
Derivative
The instantaneous rate of change of f at a point; slope of the tangent / best linear approximation when it exists.
Determinant
A scalar associated to a square matrix measuring oriented volume scaling; zero iff the matrix is singular.
Differential equation
An equation relating a function to its derivatives. Solutions are functions (or families of functions), not just numbers.
Eigenspace
For eigenvalue λ, the subspace of all eigenvectors for λ together with the zero vector; null space of A−λI.
Eigenvalue
A scalar λ such that A v = λ v for some nonzero vector v. Geometrically, a stretch factor along an invariant line.
Eigenvector
A nonzero vector sent to a scalar multiple of itself by a linear map or matrix.
Epsilon-delta definition
The precise definition of a limit: for every ε>0 there exists δ>0 such that 0<|x−a|<δ implies |f(x)−L|<ε.
Expected value
The probability-weighted average of a random variable; center of mass of its distribution.
Gradient
The vector of first partial derivatives of a scalar function; points toward steepest ascent.
Initial value problem (IVP)
A differential equation plus enough initial conditions to select a particular solution.
Integral
An antiderivative (indefinite) or a definite accumulation (definite). Linked to derivatives by the Fundamental Theorem.
Limit
The value a function or sequence approaches as the input approaches a point or infinity, when that approach settles.
Linear independence
A set of vectors is linearly independent if the only linear combination equaling zero uses all-zero coefficients.
Linear map (linear transformation)
A function between vector spaces that preserves addition and scalar multiplication.
Matrix
A rectangular array representing a linear map relative to chosen bases, or a system of linear equations.
Partial derivative
Derivative with respect to one variable while holding the others fixed.
Partial sum
The sum of the first N terms of a series; the sequence of partial sums decides convergence.
Probability
A number between 0 and 1 assigned to events under a model, satisfying nonnegativity and countable additivity (in the standard axioms).
Random variable
A numerical quantity determined by the outcome of a random experiment; described by a distribution.
Sequence
An ordered list of numbers a₁, a₂, a₃, … indexed by positive integers (or nonnegative integers).
Series
An infinite sum Σ a_n, defined as the limit of its partial sums when that limit exists.
Slope field
A plot of short segments showing the slope f(t,y) of an ODE y'=f(t,y) at sample points; guides qualitative solution shapes.
Span
The set of all linear combinations of a given collection of vectors.
Vector
An element of a vector space; informally an arrow that can be added and scaled.
Vector field
An assignment of a vector to each point of a region — e.g., velocity of a flow.