Linear span

Feb 2019
44
3
United Kingdom
Can someone clarify what it means when a spanning set of vectors spans the smallest subspace. Presumably, if the set of vectors are dependent I can find a basis set whose linear combinations will span the smallest subspace determined by the dimension. But Wiki mentions the intersection of subspaces to be the smallest subspace?
 
Aug 2012
2,496
781
Can someone clarify what it means when a spanning set of vectors spans the smallest subspace. Presumably, if the set of vectors are dependent I can find a basis set whose linear combinations will span the smallest subspace determined by the dimension. But Wiki mentions the intersection of subspaces to be the smallest subspace?
Right. If you have a set of vectors, you can generate their span as the set of all possible linear combinations of those vectors. Or, you could consider the intersection of all subspaces containing those vectors. Seemingly two different things.

Now prove they're actually the same thing.
 
Feb 2019
44
3
United Kingdom
So it’s the converse then. The intersection of all subspaces containing those vectors will give me the vectors back?

Can you give an example of a set of vectors that span multiple subspaces. I’ve always been under an illusion they can only span one. Say I have two independent vectors in space their combinations will give me a plane in space. Where do the other subspaces come from?