Hyperplane

linear-algebra

Definition

Hyperplane

A hyperplane is an affine subspace of dimension $d-1$ in a $d$-dimensional vector space ${\mathbb{R}}^d$.

It is defined by a normal vector $w\in {\mathbb{R}}^d$ and a bias term $b\in \mathbb{R}$. The set of points $x$ that form the hyperplane satisfies the linear equation:

$$H=\{x\in {\mathbb{R}}^d\mid ⟨w,x⟩+b=0\}$$

where $⟨w,x⟩$ is the inner product. The normal vector $w$ is orthogonal to every direction contained in the hyperplane, and the bias $b$ shifts the hyperplane away from the origin.

Examples

Example

In ${\mathbb{R}}^2$, a hyperplane is a line.

Example

In ${\mathbb{R}}^3$, a hyperplane is an ordinary plane. For example, with

$$w=\left(\begin{array}{c}1\\ 2\\ -1\end{array}\right),b=-3,$$

the hyperplane is

$$H=\{(x,y,z)\in {\mathbb{R}}^3\mid x+2y-z-3=0\}.$$

This is a two-dimensional plane in three-dimensional space, with normal vector $w$.