Affine Subspaces
An affine subspace (sometimes called a linear manifold, linear variety, or a flat) of a vector space is a subset closed under affine combinations of vectors in the space. For example, the set
is an affine space, where is a family of vectors in – this space is the affine span of these points. To see that this is indeed an affine space, observe that this set carries a transitive action of the vector subspace of
This affine subspace can be equivalently described as the coset of the -action
where is any element of, or equivalently as any level set of the quotient map A choice of gives a base point of and an identification of with but there is no natural choice, nor a natural identification of with
A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations. A linear subspace is an affine subspace containing the origin, or, equivalently, a subspace that is closed under linear combinations.
For example, in, the origin, lines and planes through the origin and the whole space are linear subspaces, while points, lines and planes in general as well as the whole space are the affine subspaces.
Read more about this topic: Affine Space