Hyperplane

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

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

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

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

  • The hyperplane [math]\displaystyle{ \alpha_1x_1+\alpha_2x_2+\dots+\alpha_nx_n=b }[/math] separates a space into two subspaces [math]\displaystyle{ \alpha_1x_1+\alpha_2x_2+\dots+\alpha_nx_n \lt b }[/math] and [math]\displaystyle{ \alpha_1x_1+\alpha_2x_2+\dots+\alpha_nx_n \gt b }[/math]
• Example(s):
  • Affine Hyperplane.
  • Vector Hyperplane.
  • Projective Hyperplane.
  • Support Hyperplane.
  • http://www.gabormelli.com/images/Hyperplane.gif
• Counter-example(s)
  • Vector Space,
  • Plane,
  • Subspace.
• See: Decision Boundary, Canonical Dot Product, Perceptron Algorithm, Optimal Separating Hyperplane, Line, Plane.

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References

2018

• (Mathworld, 2018) ⇒ Weisstein, Eric W. “Hyperplane." From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/Hyperplane.html Retrieved:2018-5-27
  • QUOTE: Let [math]\displaystyle{ a_1, a_2, \cdots , a_n }[/math] be scalars not all equal to 0. Then the set [math]\displaystyle{ S }[/math] consisting of all vectors

    [math]\displaystyle{ x=\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} }[/math]

    in \mathbb{R}^n</math> such that

    [math]\displaystyle{ a_1x_1+a_2x_2+\cdots +a_nx_n=c }[/math]

    for [math]\displaystyle{ c }[/math] a constant is a subspace of [math]\displaystyle{ \mathbb{R}^n }[/math] called a hyperplane.

    More generally, a hyperplane is any codimension-1 vector subspace of a vector space. Equivalently, a hyperplane V in a vector space W is any subspace such that W/V is one-dimensional. Equivalently, a hyperplane is the linear transformation kernel of any nonzero linear map from the vector space to the underlying field.

2009a

• (Wiktionary, 2009) ⇒ http://en.wiktionary.org/wiki/hyperplane#Noun
  • An n-dimensional generalization of a plane; an affine subspace of dimension n-1 that splits an n-dimensional space. (In a one-dimensional space, it is a point; In two-dimensional space it is a line; In three-dimensional space, it is an ordinary plane)

2009b

• (Wikipedia) ⇒ http://en.wikipedia.org/wiki/Hyperplane
  • In geometry, a hyperplane of an n-dimensional space [math]\displaystyle{ V }[/math] 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 [math]\displaystyle{ V }[/math] 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 [math]\displaystyle{ V }[/math] 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.

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