Affine subspace

An affine subspace of a vector space is a translate $v+W$ of a vector subspace $W$. Its dimension is the dimension of $W$.

The smallest affine subspace containing a nonempty set $S$. For any chosen $a\in S$, the displayed formula proves existence and minimality: every containing affine subspace must contain all the indicated differences in its direction space. Equivalently it consists of finite affine combinations ${\sum }_i{\lambda }_i{s}_i$ with ${\sum }_i{\lambda }_i=1$. Its dimension can be smaller than the dimension of an affine hyperplane in which the set has been embedded.

In a finite-dimensional real vector space, a level set of a nonzero linear functional is an affine hyperplane. Choosing $v_0$ in that level set gives $H=v_0+\ker \ell$, so its dimension is one less than that of the ambient space. With a Euclidean inner product, translation followed by an orthonormal basis of $\ker \ell$ identifies it isometrically with the lower-dimensional Euclidean space. The coordinate-sum constraint on balanced cut vectors is an example.

In a vector space over a field $k$, an affine line is a translate of a one-dimensional vector subspace. Its direction is that vector subspace, so multiplying $v$ by a nonzero scalar preserves the direction. Over a finite field of size $q$ it contains exactly $q$ distinct points. This usage concerns affine subsets of vector spaces, rather than the affine line as an affine scheme.