hyperplane

A hyperplane is an affine subspace of codimension one in an n-dimensional space. In Euclidean space R^n, a hyperplane can be described as the set of points x = (x1, x2, ..., xn) that satisfy a1 x1 + a2 x2 + ... + an xn = b, where the vector a = (a1, ..., an) is not the zero vector. Equivalently, it is the affine translate of the linear subspace of vectors orthogonal to a, and its normal vector a is perpendicular to every direction contained in the hyperplane. The hyperplane partitions the space into two half-spaces given by a·x ≤ b and a·x ≥ b.

Key properties include that the distance from a point x to the hyperplane is |a·x − b| divided by the norm of a, and that the hyperplane is flat and infinite in extent. If b = 0, the hyperplane passes through the origin and is a linear subspace of dimension n−1.

Examples help intuition: in R^2, a hyperplane is a line such as x + y = 1 or x = 0; in R^3, it is a plane such as 2x − y + z = 4. Hyperplanes generalize planes to higher dimensions.

Applications and related concepts vary by field. In machine learning, the decision boundary of a linear classifier is a hyperplane, separating data into classes using weights a and bias b. In linear programming and computational geometry, hyperplanes define equality constraints and influence feasible regions. The study of hyperplane arrangements investigates how several hyperplanes intersect and partition space, revealing combinatorial and geometric structure. In abstract settings, the term also extends to affine hyperplanes in vector spaces over other fields.