# finite-dimensional subspaces

• May 12th 2008, 07:12 AM
Sabita
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)}

• May 13th 2008, 02:21 PM
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.
• May 15th 2008, 03:22 AM
Sabita
Hi,
What is capW?(Wondering)
• May 15th 2008, 04:07 AM
Isomorphism
Quote:

Originally Posted by Sabita
Hi,
What is capW?(Wondering)

He means the following(though I dont know the validity of the results):

Quote:

Originally Posted by KGene
\$\displaystyle (dim U+dim W)+dim V-dim (U+V+W)\$ ........(*)
\$\displaystyle =dim (U+W)+dim (U \cap W)+dim V-dim (U+V+W)\$
\$\displaystyle =dim (U\cap W)+dim ((U+W)\cap V)\$
\$\displaystyle >=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 \$\displaystyle \mathbb{R}+\mathbb{R} = \mathbb{R}^2\$

And the validity of his idea depends on one concept,\$\displaystyle 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 :(
• May 15th 2008, 04:46 AM
KGene
Quote:

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 \$\displaystyle \mathbb{R}+\mathbb{R} = \mathbb{R}^2\$

And the validity of his idea depends on one concept,\$\displaystyle 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 \$\displaystyle U\oplus V\$. In the later case, we require \$\displaystyle U\cap V=\emptyset\$ if U,V are subspaces of a vector space. For instance, \$\displaystyle \mathbb{R}+\mathbb{R}=\mathbb{R}\$ considering \$\displaystyle \mathbb{R}\$ as a subspace of \$\displaystyle \mathbb{R}^2\$. But on the other hand, \$\displaystyle \mathbb{R}\oplus\mathbb{R}=\mathbb{R}^2\$.

2. The "formula" \$\displaystyle dim(U \oplus V) = dim(U) + dim(V) - dim(U \cap V)\$ was incorrect. The correct version is ,\$\displaystyle dim(U + V) = dim(U) + dim(V) - dim(U \cap V)\$. In general, \$\displaystyle 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.
• May 17th 2008, 07:43 AM
Sabita
Thanks KGene and Isomorphism.