1. ## 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.

2. 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! anyway, we have:

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

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

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

now (1), (2), and (3) give us the result and, much more importantly, this is my 1000 post!

3. 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!

4. 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? (just kidding!)