Proving a Vector Space Cannot be the Union of Two Proper Subspaces

[Dustinsfl]

Prove that a vector space cannot be the union of two proper
subspaces.

Let V be a vector space over a field F where U and W are proper subspaces.

I am not sure where to start with this proof.

 

[Coto]

There seems to be something missing in this problem. For example, take the case where U,W are proper subspaces of V, and U \subset W, then U \cap W \equiv W is also a proper subspace of V.

Are you sure there's not some other restrictions regarding the subspaces?

 

[Dustinsfl]

Would it matter that it is the union and you have written the intersection?

 

[Mark44]

What is the definition of "proper subspace"? You should have included that definition when you posted the problem.

 

[Dustinsfl]

What is the definition of "proper subspace"? You should have included that definition when you posted the problem.

A proper subspace can't be equal to V.

 

[Mark44]

But how is this term defined? What you gave is not the definition.

 

[Dustinsfl]

But how is this term defined? What you gave is not the definition.

If U is a proper subspace, then the dim U < dim V and U isn't the subspace of just the 0 vector, i.e., not the trivial subspaces.

 

[HallsofIvy]

So your definition of "U is a proper subspace of V" does not include requiring that U be a subset of V?

 

[vela]

There seems to be something missing in this problem. For example, take the case where U,W are proper subspaces of V, and U \subset W, then U \cap W \equiv W is also a proper subspace of V.

Are you sure there's not some other restrictions regarding the subspaces?

Would it matter that it is the union and you have written the intersection?

That was a typo. Coto meant U \cup W = W.

 

[jambaugh]

I think the issue is that the union is not the same as the span. The union of two subspaces will not be a subspace unless one of the subspaces is contained within the other (is a subspace of the subspace) in which case the union is the larger subspace.

So apply that to the question at hand...

 

[Dustinsfl]

I think the issue is that the union is not the same as the span. The union of two subspaces will not be a subspace unless one of the subspaces is contained within the other (is a subspace of the subspace) in which case the union is the larger subspace.

So apply that to the question at hand...

Thanks.