March 7th, 2019, 04:02 AM  #1 
Member Joined: Feb 2019 From: United Kingdom Posts: 44 Thanks: 3  Linear span
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?

March 7th, 2019, 07:35 AM  #2  
Senior Member Joined: Aug 2012 Posts: 2,357 Thanks: 740  Quote:
Now prove they're actually the same thing.  
March 8th, 2019, 01:17 AM  #3 
Member Joined: Feb 2019 From: United Kingdom Posts: 44 Thanks: 3 
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? 

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