Dimension Proof: U + W = dim(U) + dim(W) - dim(U ∩ W)

[cookiesyum]

Homework Statement

Let U, W be subspaces of a vector space V. Show that dim(U+W) = dim(U) + dim (W) - dim (U intersect W).

Homework Equations

The Attempt at a Solution

I can see this picture-wise in a venn-diagram form. In adding U and W you count the elements in their intersection twice, since both spaces contain them. Thus, you subtract the dimension of U intersect W. But, how do you show this in proof format? It doesn't seem like a verbal explanation is enough.

 

[Dick]

What you really are thinking about in your Venn diagram are basis vectors. Write a basis for U intersect W. Extend it to a basis for U and W. Now do the Venn argument.