Let K {\displaystyle {}K} be a field, and let V {\displaystyle {}V} be a K {\displaystyle {}K} -vector space. Let v i {\displaystyle {}v_{i}} , i ∈ I {\displaystyle {}i\in I} , be a family of vectors in V {\displaystyle {}V} and w j {\displaystyle {}w_{j}} , j ∈ J {\displaystyle {}j\in J} , another family of vectors in V {\displaystyle {}V} . Then, for the spanned linear subspaces, the inclusion ⟨ v i , i ∈ I ⟩ ⊆ ⟨ w j , j ∈ J ⟩ {\displaystyle {}\langle v_{i},\,i\in I\rangle \subseteq \langle w_{j},\,j\in J\rangle } holds, if and only if v i ∈ ⟨ w j , j ∈ J ⟩ {\displaystyle {}v_{i}\in \langle w_{j},\,j\in J\rangle } holds for all i ∈ I {\displaystyle {}i\in I} .
be a
field,
and let 
be a
-vector space.
Let
, 
,
be a family of vectors in and
, 
,
another family of vectors in . Then, for the
spanned linear subspaces,
the inclusion
holds, if and only if
holds for all
.
