For every , where is the smallest
cone containing (here, a set is a cone if for all and all non-negative ).
Hence is always a linear subspace of parallel to .
Related sets
If instead of an affine combination one uses a
convex combination, that is, one requires in the formula above that all be non-negative, one obtains the
convex hull of S, which cannot be larger than the affine hull of S, as more restrictions are involved.
If however one puts no restrictions at all on the numbers , instead of an affine combination one has a
linear combination, and the resulting set is the
linear span of S, which contains the affine hull of S.
For every , where is the smallest
cone containing (here, a set is a cone if for all and all non-negative ).
Hence is always a linear subspace of parallel to .
Related sets
If instead of an affine combination one uses a
convex combination, that is, one requires in the formula above that all be non-negative, one obtains the
convex hull of S, which cannot be larger than the affine hull of S, as more restrictions are involved.
If however one puts no restrictions at all on the numbers , instead of an affine combination one has a
linear combination, and the resulting set is the
linear span of S, which contains the affine hull of S.