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).
