Dimension Proof?

1. The problem statement, all variables and given/known data
Let U, W be subspaces of a vector space V. Show that dim(U+W) = dim(U) + dim (W) - dim (U intersect W).

2. Relevant equations

3. 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.


Science Advisor
Homework Helper
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.

The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving