Let
,
endowed with the
standard inner product.
For the one-dimensional
linear subspace
, generated by the standard vector , the
orthogonal complement
consits of all vectors , where the -th entry is . For a one-dimensional linear subspace , generated by a vector
-
the orthogonal complement can be found by determining the solution space of the
linear equation
-
The orthogonal space
-
has dimension ; this is a so-called
hyperplane.
The vector is called a normal vector for the hyperplane .
For a linear subspace
that is given by a
basis
(or a
generating system)
, ,
the orthogonal complement is the solution space of the
system of linear equations
-
where
is the matrix formed by the vectors as rows.