# Lin. Algebra - Sum of Dim. of Three Subspaces

1. Feb 21, 2008

### steelphantom

Another linear algebra question! What a surprise!

1. The problem statement, all variables and given/known data
If U1, U2, U3, are subspaces of a finite-dimensional vector space, then show

dim(U1 + U2 + U3) = dimU1 + dimU2 + dimU3 - dim(U1 $$\cap$$ U2) - dim(U1 $$\cap$$ U3) - dim(U2 $$\cap$$ U3) + dim(U1 $$\cap$$ U2 $$\cap$$ U3)

or give a counterexample.

2. Relevant equations

3. The attempt at a solution
I have the proof of the sum of the dimension of two subspaces in my book, so I would assume I would proceed in much the same way, but that "or give a counterexample" is making me just a little bit uneasy. I'm 90% sure that this is true, because basically the same formula holds for sets. Could anyone tell me if this is true before I proceed with my proof? It's gonna be a long one if I use the same method the book did.

2. Feb 21, 2008

### Dick

This has been asked more than once today, and I'll give you the same answer others have given. It is the same formula as for sets. And for the same reasons if you pick a compatible basis for the vector space. Proceed with your proof.

3. Feb 22, 2008

### steelphantom

Thanks for the response. I finished my proof, but where was this question asked earlier today? I didn't see it in the Homework Help or Linear Algebra forums.

4. Feb 22, 2008

### Dick

Hmmm. Now I can't find it. It may be under an obscure title. BTW, I didn't mean to say that you should have searched for other posts before asking. I was only saying my response wasn't original.