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

Another linear algebra question! What a surprise! ## Homework Statement

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.

## 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.

## Answers and Replies

Dick
Science Advisor
Homework Helper
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.

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.

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.

Dick
Science Advisor
Homework Helper
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.

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.