1. ## finite-dimensional subspaces

Hi,
Have been trying to understand linear Algebra for a month now. Am unable to solve this:
If U,V,W are finite-dimensional subspaces of a real vector space, show that
dim U + dim V + dim W - dim(U+V+W) >= max{dim(U intersection V), dim(V intersection W), dim(U intersection W)}

2. (dim U+dim W)+dim V-dim (U+V+W) ........(*)
=dim (U+W)+dim (U\cap W)+dim V-dim (U+V+W)
=dim (U\cap W)+dim ((U+W)\cap V)
>=dim (U\cap W)

Since U,V,W are independent, and (*) is symmetric, so you get three inequalities.

3. Hi,
What is capW?

4. Originally Posted by Sabita
Hi,
What is capW?
He means the following(though I dont know the validity of the results):

Originally Posted by KGene
$(dim U+dim W)+dim V-dim (U+V+W)$ ........(*)
$=dim (U+W)+dim (U \cap W)+dim V-dim (U+V+W)$
$=dim (U\cap W)+dim ((U+W)\cap V)$
$>=dim (U\cap W)$

Since U,V,W are independent, and (*) is symmetric, so you get three inequalities.
Actually I dont know what U+V means. Is it the direct sum?
Like $\mathbb{R}+\mathbb{R} = \mathbb{R}^2$

And the validity of his idea depends on one concept, $dim(U \oplus V) = dim(U) + dim(V) - dim(U \cap V)$
But wiki says "The dimension of V ⊕ U is equal to the sum of the dimensions of V and U"...
So I am not sure

5. Originally Posted by Isomorphism
He means the following(though I dont know the validity of the results):

Actually I dont know what U+V means. Is it the direct sum?
Like $\mathbb{R}+\mathbb{R} = \mathbb{R}^2$

And the validity of his idea depends on one concept, $dim(U \oplus V) = dim(U) + dim(V) - dim(U \cap V)$
But wiki says "The dimension of V ⊕ U is equal to the sum of the dimensions of V and U"...
So I am not sure
1. The sum U+V means the vector space consisting of elements of the form v+w where v in V and w in W, it is different from the direct sum $U\oplus V$. In the later case, we require $U\cap V=\emptyset$ if U,V are subspaces of a vector space. For instance, $\mathbb{R}+\mathbb{R}=\mathbb{R}$ considering $\mathbb{R}$ as a subspace of $\mathbb{R}^2$. But on the other hand, $\mathbb{R}\oplus\mathbb{R}=\mathbb{R}^2$.

2. The "formula" $dim(U \oplus V) = dim(U) + dim(V) - dim(U \cap V)$ was incorrect. The correct version is , $dim(U + V) = dim(U) + dim(V) - dim(U \cap V)$. In general, $dim (U\oplus V)=dim U+dim V$

3. In my first post, I meant the sum of two vector subspaces U+V, not the direct sum.

Hope these help.

6. Thanks KGene and Isomorphism.