Parallel Affine Set - Knowlpedia

Let $X$ be a vector space. An affine set $\Omega\subset X$ is said to be parallel to a linear subspace $L\subset X$ if

\Omega=\omega+L

for some $\omega\in\Omega$ .

Examples

In $\mathbb{R}^n$ , any affine subspace is parallel to its direction subspace (the translation of the affine set through the origin).

Remarks

By the translate characterization, every nonempty affine set is parallel to at least one subspace. The parallel subspace is unique and can be written explicitly as $\Omega-\Omega$ ; see the Ω−Ω proposition.