Affine subspace

Definition

Let $K$ be a field, let $V$ be a vector space over $K$, and let $A$ be an affine space over $V$. A nonempty subset $S$ of $A$ is an affine subspace if $S = p + W$ for a point $p$ of $A$ and a linear subspace $W$ of $V$. This means that $$S = p + W = \{\, p + w \;:\; w \in W \,\}.$$ The subspace $W$ is the direction of $S$. It is unique: $W = \{\, q - r : q, r \in S \,\}$. The dimension of $S$ is the dimension of $W$.

Remarks

For each point $p$ of $S$, $S = p + W$. Thus each point of $S$ can serve as the base point. In $\mathbb{R}^n$, the affine subspaces of dimension 1 are the lines, and those of dimension 2 are the planes. Those of dimension $n - 1$ are the hyperplanes. Two affine subspaces are parallel if they have the same direction. Mathlib's AffineSubspace also allows the empty set. A subset $S$ is an affine subspace in this sense if $c(q - r) + s \in S$ for all $c \in K$ and $q, r, s \in S$.

Explanations

L1 human

An affine subspace is a flat piece of space that need not pass through the origin. A line or a plane that has been moved away from the origin is an example. It is a linear subspace that has been shifted by a translation.