Proposition 2

For a non-empty subset $V$ of an affine space $E$ to be a linear variety, it is necessary and sufficient that $V$ be a vector subspace for the vector space structure on $E$ obtained by taking a point of $V$ as origin.

In particular, the non-empty affine linear varieties of a vector space $T$ (considered as an affine space) are just the translates of the vector subspaces of $T$; the vector subspaces of $T$ are therefore the linear varieties containing 0.

Let $V$ be a non-empty linear variety of the affine space $E$; the set of free vectors $x - y$, where $x$ and $y$ run through $V$, is a vector subspace $D$ of the translation space $T$ of $E$ called the direction of $V$: for, if $a \in V$, then

and our assertion follows from Proposition 2. It is immediate that $D$ operates faithfully and transitively on $V$, which therefore has canonically the structure of an affine space attached to $D$. By the dimension of the affine variety $V$, we mean the dimension of $V$ with this affine space structure, that is the dimension of the vector space $D$. The linear varieties of dimension 0 are the points of $E$; those of dimension 1 (resp. 2) are called lines (resp. planes) of $E$.

Every vector $\neq 0$ belonging to the direction of a line is called a direction vector of this line; its components with respect to a basis of T form what is called a system of direction parameters of the line in question.

The codimension of a linear variety V in E is the codimension of its direction D in T; a linear variety of codimension 1 in E is called an (affine) hyperplane of E.

Two linear varieties with the same direction are called parallel; it amounts to the same to say that one is derived from the other by translation. If V is a linear variety in T (considered as an affine space), its direction is the linear variety parallel to V and containing 0.