# Union of two subspaces

1. Aug 25, 2011

### 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.

2. Aug 25, 2011

### 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?

3. Aug 25, 2011

### Dustinsfl

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

4. Aug 26, 2011

### Staff: Mentor

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

5. Aug 26, 2011

### Dustinsfl

A proper subspace can't be equal to V.

6. Aug 26, 2011

### Staff: Mentor

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

7. Aug 26, 2011

### Dustinsfl

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.

8. Aug 26, 2011

### HallsofIvy

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

9. Aug 26, 2011

### vela

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

10. Aug 26, 2011

### 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...

11. Aug 26, 2011

Thanks.