# Finite-dimensional subspaces..?

• May 5th 2009, 05:51 AM
fardeen_gen
Finite-dimensional subspaces..?
Let U, V, W be finite-dimensional subspaces of a real vector space. Show dim U + dim V + dim W - dim (U + V + W) greater or equal to max {dim (U intersection V), dim (V intersection W), dim (U intersection W)}.

- asked on behalf of a friend who is a non MHFer.

I personally have no idea about vector spaces.
• May 5th 2009, 06:17 AM
NonCommAlg
Quote:

Originally Posted by fardeen_gen
Let U, V, W be finite-dimensional subspaces of a real vector space. Show dim U + dim V + dim W - dim (U + V + W) greater or equal to max {dim (U intersection V), dim (V intersection W), dim (U intersection W)}.

- asked on behalf of a friend who is a non MHFer.

I personally have no idea about vector spaces.

i hope your friend is better than you at linear algebra! (Wink) anyway, we have:

$\dim(U \cap V) + \dim ((U+V)\cap W)=\dim U+ \dim V + \dim W -\dim(U+V+W).$ thus: $\dim(U \cap V) \leq \dim U + \dim V + \dim W -\dim(U+V+W), \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$

and: $\dim(U \cap W) + \dim ((U+W)\cap V)=\dim U+ \dim V + \dim W -\dim(U+V+W).$ thus: $\dim(U \cap W) \leq \dim U + \dim V + \dim W -\dim(U+V+W), \ \ \ \ \ \ (2)$

and: $\dim(V \cap W) + \dim ((V+W) \cap U)=\dim U+ \dim V + \dim W -\dim(U+V+W).$ thus: $\dim(V \cap W) \leq \dim U + \dim V + \dim W -\dim(U+V+W). \ \ \ \ \ \ \ (3)$

now (1), (2), and (3) give us the result and, much more importantly, this is my 1000 post! (Party) (Rofl)
• May 5th 2009, 08:32 AM
fardeen_gen
(Clapping) Congratulations! Eventful journey on MHF indeed.

P.S. My friend is 24 yrs old. I am a 16 yr old :) I have no idea what linear algebra is!
• May 5th 2009, 09:19 AM
NonCommAlg
Quote:

Originally Posted by fardeen_gen

P.S. My friend is 24 yrs old. I am a 16 yr old :) I have no idea what linear algebra is!

wow, you're just a kid! i think it's quite late in India now. shouldn't you be in bed? (Smile) (just kidding!)