Let K {\displaystyle {}K} be a field, let V {\displaystyle {}V} be a K {\displaystyle {}K} -vector space, and let U ⊆ V {\displaystyle {}U\subseteq V} denote a linear subspace. Let u i {\displaystyle {}u_{i}} , i ∈ I {\displaystyle {}i\in I} , denote a basis of U {\displaystyle {}U} , and v j {\displaystyle {}v_{j}} , j ∈ J {\displaystyle {}j\in J} , a family of vectors in V {\displaystyle {}V} . Show that the family u i , i ∈ I , v j , j ∈ J {\displaystyle {}u_{i},i\in I,v_{j},j\in J} , is a basis of V {\displaystyle {}V} if and only if [ v j ] {\displaystyle {}[v_{j}]} , j ∈ J {\displaystyle {}j\in J} , is a basis of the residue class space V / U {\displaystyle {}V/U} .
be a
field,
let 
be a
-vector space,
and let
denote a
linear subspace. Let
, 
,
denote a
basis
of 
, and
, 
,
a family of vectors in . Show that the family 
, is a basis of if and only if
![{\displaystyle {}[v_{j}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3de511c2c34a7b306d1459100b6101b424293cf3)
, ,
is a basis of the
residue class space
.
