# Hyperplane

(Redirected from hyperplane)

A hyperplane, $\displaystyle{ H }$, is a k-dimensional structure that divides a k+1 dimensional space into two space regions.

• AKA: H.
• Context:
• It can have distance to the Origin of $\displaystyle{ -b/\|w\| }$.
• It can have distance from an arbitrary point x (to the hyperplane) of ((w·x)+b)/ ||w||
• If $\displaystyle{ x \in H }$ then $\displaystyle{ (w · x)+b=0 }$.
• For $\displaystyle{ \alpha_1,\alpha_2,\dots,\alpha_n }$ be scalars not all equal to zero and the set $\displaystyle{ S }$ consisting of all vectors $\displaystyle{ X=\begin{bmatrix} x_1 \\ x_2 \\ \vdots\\ x_n \end{bmatrix} \in \mathbb{R} }$, the equation $\displaystyle{ \alpha_1x_1+\alpha_2x_2+\dots+\alpha_nx_n=b }$ for $\displaystyle{ b }$ a constant is a subspace of $\displaystyle{ \mathbb{R}^n }$ is called a hyperplane.
• If a space is of 2-dimentional $\displaystyle{ \mathbb{R}^2 }$ then the hyperplanes are 1-dimentional lines.

Here $\displaystyle{ \alpha_1x_1+\alpha_2x_2=b }$ the line, is the equation of the hyperplane.$\displaystyle{ \alpha_1x_1+\alpha_2x_2 \lt b }$ and $\displaystyle{ \alpha_1x_1+\alpha_2x_2 \gt b }$ are the two subspaces separated by the hyperplane.

• If a space is of 3-dimentional $\displaystyle{ \mathbb{R}^3 }$ then the hyperplanes are 2-dimentional planes.

Here $\displaystyle{ \alpha_1x_1+\alpha_2x_2+\alpha_3x_3=b }$ the plane, is the equation of the hyperplane.$\displaystyle{ \alpha_1x_1+\alpha_2x_2+\alpha_3x_3 \lt b }$ and $\displaystyle{ \alpha_1x_1+\alpha_2x_2+\alpha_3x_3 \gt b }$ are the two subspaces separated by the hyperplane.

• The hyperplane $\displaystyle{ \alpha_1x_1+\alpha_2x_2+\dots+\alpha_nx_n=b }$ separates a space into two subspaces $\displaystyle{ \alpha_1x_1+\alpha_2x_2+\dots+\alpha_nx_n \lt b }$ and $\displaystyle{ \alpha_1x_1+\alpha_2x_2+\dots+\alpha_nx_n \gt b }$
• Example(s):
• Counter-example(s)
• See: Decision Boundary, Canonical Dot Product, Perceptron Algorithm, Optimal Separating Hyperplane, Line, Plane.

## References

### 2009b

• (Wikipedia) ⇒ http://en.wikipedia.org/wiki/Hyperplane
• In geometry, a hyperplane of an n-dimensional space $\displaystyle{ V }$ is a "flat" subset of dimension n − 1, or equivalently, of codimension 1 in V ; it may therefore be referred to as an (n − 1)-flat of V. The space $\displaystyle{ V }$ may be an Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly; in all cases however, any hyperplane can be given in coordinates as the solution of a single (because of "codimension 1") algebraic equation of degree 1 (because of "flat"). If $\displaystyle{ V }$ is a vector space, one distinguishes "vector hyperplanes" (which are subspaces, and therefore must pass through the origin) and "affine hyperplanes" (which need not pass though the origin; they can be obtained by translation of a vector hyperplane). A hyperplane in a Euclidean space separates that space into two half spaces, and defines a reflection that fixes the hyperplane and interchanges those two half spaces.