A hyperplane in 
is the set of solutions of a single linear
equation with coefficients not all equal to 0.
More explicitly, it consists of all vectors
![X=[x_1; x_2; |; x_n]](/images/equations/Hyperplane/NumberedEquation1.svg) |
such that
 |
for fixed scalars 
, 
,
..., 
not all equal to 0 and a constant 
. This is an affine
space of dimension 
. It is a vector subspace of
iff 
.
More generally, a linear hyperplane is a codimension-1 subspace 
of a vector space 
. Equivalently, the quotient
vector space 
is one-dimensional, or 
is the linear transformation kernel
of a nonzero linear
functional on 
.
A linear hyperplane is also called a hyperspace (Hoffman
and Kunze 1971, pp. 101, 109-110). An affine hyperplane is obtained by applying
a translation to a linear hyperplane and need not
contain the origin.
Two nonzero linear functionals have the same linear transformation kernel iff one is a nonzero scalar multiple of the other (Hoffman and Kunze 1971, p. 110).
