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Let a_1, a_2, ..., a_n be scalars not all equal to 0. Then the set S consisting of all vectors

X=[x_1; x_2; |; x_n]

in R^n such that

a_1x_1+a_2x_2+...+a_nx_n=c

for c a constant is a subspace of R^n 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.

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Plane, Subspace Explore this topic in the MathWorld classroom

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Cite this as:

Weisstein, Eric W. "Hyperplane." FromMathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Hyperplane.html

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