is a subspace
if every linear combination of points
is also in S
is an affine combination
(Note that we do not require
A set M is affine if every affine combination of points in M
is also in M.
are affinely independent
if the vectors
are linearly independent.
Given a scalar
and a vector
is a halfspace
A polyhedron is a finite intersection of halfspaces.
Note that the feasible region of a linear programming problem is a
The dimension of a subspace is the maximum number of
linearly independent vectors in it.
Every affine space is a translation of a subspace.
Further, the subspace is uniquely defined by the affine space.
The dimension of an affine space
is the dimension of the corresponding subspace.
The affine hull of a set is the set of all affine combinations
of points in the set.
This is equivalent to the intersection of all affine sets containing the
The dimension of a polyhedron is the dimension of its affine hull.
be a polyhedron.
be the hyperplane
is a face
Let P be a polyhedron of dimension d.
A face of dimension d-1 is a facet.
A face of dimension 1 is an edge.
A face of dimension 0 is a vertex.
John E Mitchell