Let
be a field and let
be a
-vector space. Prove the following facts.
- Let
,
,
be a family of subspaces of
. Prove that also the intersection
-
![{\displaystyle {}U=\bigcap _{j\in J}U_{j}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/884d8c6452e4bcc98c72604b9de56bed8bfb8e4b)
is a subspace.
- Let
,
,
be a family of elements of
and consider the subset
of
which is given by all linear combinations of these elements. Show that
is a subspace of
.
- The family
,
,
is a system of generators of
if and only if
-
![{\displaystyle {}\langle v_{i},\,i\in I\rangle =V\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afd8a63388e1a75b62696ce79ed5c172e38fa80e)