Vector space/Generating system and spanning subspace/Fact/Proof/Exercise
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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
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