What is affine space?

Example 17.2 (An affine space with no natural origin)

The solution plane $\mathcal E = \{(x, y, z) \in \R^3 : x + y + z = 1\}$ is not a vector subspace ($0 \notin \mathcal E$), but it is an affine space directed by $E = \{x + y + z = 0\}$: for $A, B \in \mathcal E$ the difference $\vect{AB} = B - A$ lands in $E$ (the sums cancel), Chasles is inherited from $\R^3$, and $B \mapsto \vect{AB}$ is bijective onto $E$. No point of $\mathcal E$ is distinguished — any choice of “origin” $O \in \mathcal E$ works equally well, and all the identifications $M \mapsto \vect{OM}$ differ by translations. This is the typical situation: solution sets of inhomogeneous linear problems (linear systems, linear differential equations in Chapter 16) are affine, never linear, and the slogan “particular solution plus kernel” is exactly the statement $\mathcal F = A + F$ of the next definition.