The affine subspaces of 3-dimensional Euclidean space are the points, lines, planes, and whole space. The translations of these subspaces are all vector spaces. Formallly an affine space is a map S × X → S where S is a set and X is a vector space X, and where we write s + x for the image of (s, x), satisfying

1. (i) s + 0 = s,

2. (ii) (s + x) + y = s + (x + y).

3. (iii) given s, t ∈ S, there is a unique x ∈ X such that s + x = t.

The solutions of an inhomogeneous linear equation form an affine space. For example, the solutions to the ODE y”−y = 1 has a solution set S and the vector space X is the solution space of y”−y = 0. Any vector space V is an affine space with S = X = V.