Editing Hyperplane (section)

=== Affine hyperplanes ===

An '''affine hyperplane''' is  an [[affine space|affine subspace]] of [[codimension]] 1 in an [[affine space]].
In [[Cartesian coordinates]], such a hyperplane can be described with a single [[linear equation]] of the following form (where at least one of the <math>a_i</math>s is non-zero and <math>b</math> is an arbitrary constant):

:<math>a_1x_1 + a_2x_2  + \cdots + a_nx_n = b.\ </math>

In the case of a real affine space, in other words when the coordinates are real numbers, this affine space separates the space into two half-spaces, which are the [[connected component (topology)|connected component]]s of the [[complement (set theory)|complement]] of the hyperplane, and are given by the [[inequality (mathematics)|inequalities]]

:<math>a_1x_1 + a_2x_2  + \cdots + a_nx_n < b\ </math>

and

:<math>a_1x_1 + a_2x_2  + \cdots + a_nx_n > b.\ </math>

As an example, a point is a hyperplane in 1-dimensional space, a line is a hyperplane in 2-dimensional space, and a plane is a hyperplane in 3-dimensional space. A line in 3-dimensional space is not a hyperplane, and does not separate the space into two parts (the complement of such a line is connected).

Any hyperplane of a Euclidean space has exactly two unit normal vectors: <math>\pm\hat{n}</math>. In particular, if we consider <math>\mathbb{R}^{n+1}</math> equipped with the conventional inner product ([[dot product]]), then one can define the affine subspace with normal vector <math>\hat{n}</math> and origin translation <math>\tilde{b} \in \mathbb{R}^{n+1}</math> as the set of all <math>x \in \mathbb{R}^{n+1}</math> such that <math>\hat{n} \cdot (x-\tilde{b})=0</math>.

Affine hyperplanes are used to define decision boundaries in many [[machine learning]] algorithms such as linear-combination (oblique) [[Decision tree learning|decision trees]], and [[perceptron]]s.