stylistic:style_transfer
Subspaces

Let E be a Euclidean space and E → {\displaystyle {\overrightarrow {E}}} its associated vector space.

A flat, Euclidean subspace or affine subspace of E is a subset F of E such that

F → = { P Q → | P ∈ F, Q ∈ F } ({\displaystyle {\overrightarrow {F}}={\Bigl \{}{\overrightarrow {PQ}}\mathrel {\Big |} P\in F,Q\in F{\Bigr \}}{\vphantom {\frac {(}{}}}}

as the associated vector space of F is a linear subspace (vector subspace) of E →. {\displaystyle {\overrightarrow {E}}.} A Euclidean subspace F is a Euclidean space with F → {\displaystyle {\overrightarrow {F}}} as the associated vector space. This linear subspace F → {\displaystyle {\overrightarrow {F}}} is also called the direction of F.

If P is a point of F then

F = { P + v | v ∈ F → }. {\displaystyle F={\Bigl \{}P+v\mathrel {\Big |} v\in {\overrightarrow {F}}{\Bigr \}}.}

Conversely, if P is a point of E and V → {\displaystyle {\overrightarrow {V}}} is a linear subspace of E →, {\displaystyle {\overrightarrow {E}},} then

P + V → = { P + v | v ∈ V → } {\displaystyle P+{\overrightarrow {V}}={\Bigl \{}P+v\mathrel {\Big |} v\in {\overrightarrow {V}}{\Bigr \}}}

is a Euclidean subspace of direction V → {\displaystyle {\overrightarrow {V}}}. (The associated vector space of this subspace is V → {\displaystyle {\overrightarrow {V}}}.)

A Euclidean vector space E → {\displaystyle {\overrightarrow {E}}} (that is, a Euclidean space that is equal to E → {\displaystyle {\overrightarrow {E}}}) has two sorts of subspaces: its Euclidean subspaces and its linear subspaces. Linear subspaces are Euclidean subspaces and a Euclidean subspace is a linear subspace if and only if it contains the zero vector.