TITLE: Is there a version of inclusion/exclusion for vector spaces?
QUESTION [31 upvotes]: I am asking for a way to compute the rank of the 'join' of a bunch of subspaces whose pairwise intersections might be non-zero. So in the case n=2 this is just $\dim(A_1+A_2) = \dim(A_1) + \dim(A_2) - \dim(A_1\cap A_2)$. For general $n$, I don't know a good formula.

REPLY [5 votes]: I just wanted to point out, for other readers, the excellent paper by Gian-Carlo Rota "On the foundations of combinatorial theory I. Theory of Möbius function"
http://www.maths.ed.ac.uk/~aar/papers/rota1.pdf
The paper somehow sets up the proper framework to generalize the inclusion-exclusion principle (which is basically a special case of Möbius inversion).
(I wanted to make a comment following Shor's answer, but it seems I don't have enough reputation credit ^^ ...)

REPLY [2 votes]: The question immediately reminded me of this answer to the famous post of "Examples of common false beliefs in mathematics". Obviously, it does not answer the question, but I believe it adds something informative and worthy.