\[\mathbb{R}^n\]
be the set of n - tuples. A k - flat \[F\]
, or linear variety or affine subspace, in \[\mathbb{R}^n\]
is the set of all n - tuples of the form \[P+V\]
, where \[P\]
is the position vector of a point of \[\mathbb{R}^n\]
and \[\mathbf{w}\]
is from a a k - dimensional subspace of \[\mathbb{R}^n\]
.\[W\]
is called a direction space of \[F\]
and we can write \[F=P+W\]
.A point is a 0 - flat (ha dimension 0).
A line is a 1 - flat (ha dimension 1).
A plane is a 2 - flat (ha dimension 2).