Affine space
Affine subspaces and parallelism

An affine subspace (also called, in some contexts, a linear variety, a flat, or, over the real numbers, a linear manifold) B of an affine space A is a subset of A such that, given a point a ∈ B {\displaystyle a\in B}, the set of vectors B → = { b − a ∣ b ∈ B } {\displaystyle {\overrightarrow {B}}=\{b-a\mid b\in B\}} is a linear subspace of A → {\displaystyle {\overrightarrow {A}}}. This property, which does not depend on the choice of a, implies that B is an affine space, which has B → {\displaystyle {\overrightarrow {B}}} as its associated vector space.

The affine subspaces of A are the subsets of A of the form

where a is a point of A, and V a linear subspace of A → {\displaystyle {\overrightarrow {A}}}.

The linear subspace associated with an affine subspace is often called its direction, and two subspaces that share the same direction are said to be parallel.

This implies the following generalization of Playfair's axiom: Given a direction V, for any point a of A there is one and only one affine subspace of direction V, which passes through a, namely the subspace a + V.

Every translation A → A: a ↦ a + v {\displaystyle A\to A:a\mapsto a+v} maps any affine subspace to a parallel subspace.

The term parallel is also used for two affine subspaces such that the direction of one is included in the direction of the other.
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