Hyperplane

A hyperplane is a flat subspace whose dimension is one less than that of its ambient space. In two dimensions it is a line; in three dimensions, a plane; and in one dimension, a point. The precise definition depends on whether the ambient space is a vector space, an affine space, or a projective space. Hyperplanes provide a common language for linear equations, geometric separation, and classification. (ee263.stanford.edu)

Definition and coordinate representation

In the real coordinate space $\mathbb{R}^n$, an affine hyperplane has the form

where $a=(a_1,\ldots,a_n)$ is a coefficient vector and $b$ is a scalar. Equivalently, its defining equation is

The nonzero-coefficient condition is essential: if $a=0$, the equation defines either the whole space or the empty set, rather than a hyperplane. Multiplying both $a$ and $b$ by the same nonzero scalar leaves the set unchanged. Thus, a hyperplane does not have a unique defining equation. (ee263.stanford.edu)

When $b=0$, the hyperplane contains the origin and is a linear subspace, often called a linear or vector hyperplane. For arbitrary $b$, choosing any $x_0\in H$ gives

An affine hyperplane is therefore a translate of a linear hyperplane. It need not be closed under vector addition or scalar multiplication. Both types are affine sets and convex sets. (stanford.edu)

Algebraic characterization

The coordinate-free definition uses a nonzero linear map $f:V\to K$, where $K$ is the underlying field. Such a scalar-valued map is called a linear functional. A linear hyperplane is exactly a set

Conversely, every linear subspace of codimension one is the kernel of a nonzero linear functional. The functionals defining the same hyperplane differ only by a nonzero scalar factor. (people.clas.ufl.edu)

For finite-dimensional $V$, the rank–nullity theorem explains the dimension: a nonzero functional has a one-dimensional image, so its kernel has dimension $\dim V-1$. More generally, codimension one means that the quotient space $V/H$ has dimension one. This formulation also applies to infinite-dimensional vector spaces. The relevant functionals belong to the dual space $V^*$. (people.clas.ufl.edu)

Hyperplanes also describe individual constraints in a system of linear equations. For a consistent system $Ax=c$, with $n$ unknowns and matrix rank $r$, the solution set is an affine subspace of dimension $n-r$. Several hyperplane constraints therefore need not produce another hyperplane: independent constraints reduce dimension further. (ee263.stanford.edu)

Normal vectors, distance, and projection

In real Euclidean space, $a$ is perpendicular to every direction lying within $H$, and is called a normal vector. Indeed, differences of points on $H$ satisfy $a^\mathsf{T}(x-y)=0$. This interpretation uses the Euclidean inner product; the algebraic definition itself requires no notion of perpendicularity. (ee263.stanford.edu)

The distance from a point $p$ to $H$ is

Removing the absolute value gives an oriented signed distance. Its sign indicates the side of the hyperplane containing $p$, once an orientation has been chosen. The nearest point, or orthogonal projection, is

For example, the plane $x+2y+2z=3$ has normal vector $(1,2,2)$; substitution into the distance formula shows that its distance from the origin is $1$. (cs.cmu.edu)

Half-spaces and convex geometry

A real affine hyperplane divides $\mathbb{R}^n$ into two open half-spaces, specified by $a^\mathsf{T}x<b$ and $a^\mathsf{T}x>b$. Replacing strict inequalities by non-strict ones produces closed half-spaces sharing $H$ as their boundary. This description depends on the ordering of real numbers and is not an automatic property of hyperplanes over arbitrary fields. (stanford.edu)

A supporting hyperplane meets a convex set while placing the entire set in one closed half-space. A separating hyperplane places two sets on opposite sides, possibly allowing contact with the boundary. The separating hyperplane theorem guarantees weak separation for two nonempty, disjoint convex sets in finite-dimensional real space; strict separation requires additional hypotheses. (stanford.edu)

In linear programming, affine inequalities define half-spaces whose intersection forms the feasible set. Level sets of a nonconstant linear objective function are parallel hyperplanes. At an attained optimum, the corresponding level hyperplane supports the feasible set, giving a geometric interpretation of optimization. (web.stanford.edu)

Machine learning

In machine learning, a classifier can assign labels according to the sign of $w^\mathsf{T}x+\beta$. Its decision boundary is a hyperplane when $w\ne0$. Two labeled collections have linear separability when such a boundary places them strictly on opposite sides. (cs229.stanford.edu)

A hard-margin support vector machine selects a separating hyperplane maximizing the minimum distance to the training data. With labels $y_i\in\{-1,1\}$, its standard constraints are $y_i(w^\mathsf{T}x_i+\beta)\ge1$, and it minimizes $\tfrac12\|w\|_2^2$. Soft-margin variants allow violations. With a kernel method, the separating hyperplane lies in a transformed feature space, while its boundary in the original input space may be nonlinear. (cs229.stanford.edu)

Projective hyperplanes

In projective geometry, a hyperplane in $\mathbb{P}^n(K)$ has projective dimension $n-1$. In homogeneous coordinates it is defined by

with coefficients not all zero. Because homogeneous coordinates represent points only up to nonzero scaling, the equation is unchanged by rescaling a representative. Projective hyperplanes correspond to points of the projective space associated with the dual vector space—a basic instance of projective duality. (cis.upenn.edu)