Geometrically, an affine subset is something that looks like a subspace except that it doesn't contain the origin. Algebraically, if S is a subspace and v is a vector, then the set v+S (in other words, the set of all vectors of the form v plus something in S) is an affine set, and every affine set is of this form.

In the above example, start by finding a vector in M. The easiest one to find is probably . Now let . Show that S is a subspace of , and M = v + S. Therefore M is an affine set.

An affine set is simply a subspace that has been translated away from the origin.