Hyperplane

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

      Here [math]\alpha_1x_1+\alpha_2x_2=b[/math] the line, is the equation of the hyperplane.[math]\alpha_1x_1+\alpha_2x_2 \lt b[/math] and [math]\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]\mathbb{R}^3[/math] then the hyperplanes are 2-dimentional planes.

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

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


References

2018

2009a

2009b

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