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.
[Math] version of inclusion/exclusion for vector spaces
linear algebra
Best Answer
As the blog link points out, for any finite collection of subspaces U_1,...,U_n there's a chain complex
$$0 \to \cap U_i \to \ldots \to \bigoplus U_i \cap U_j \cap U_k \to \bigoplus U_i \cap U_j \to \bigoplus U_i \to \sum U_i \to 0$$
where the rth term after the left hand zero is the external direct sum of (n-r+1)-fold intersections of U_i's. The differential sends $$x \in U_{i_1} \cap \cdots \cap U_{i_r}$$ to $$\sum_j (-1)^j (x \in U_{i_1} \cap \cdots \hat{U_{i_j}} \cap \cdots \cap U_{i_r})$$
Failure of "inclusion-exclusion for vector spaces" is failure of exactness of this sequence. For example, for three subspaces the only non-trivial homology is H^1 which is $(U \cap (V+W))/(U\cap V + U \cap W)$ i.e. it measures failure of distributivity.
You can find out what the Euler characteristic of the sequence is by repeatedly using the formula for dim(U+V). What this gives for 4 subspaces is that the alternating sum of Betti numbers is
$$|((U \cap V) \cap (U\cap W + U \cap X)) / (U \cap V \cap W + U \cap V \cap X) |$$ minus the sum of $$| (V \cap (W+X))/(V \cap W + V \cap X) |$$ and $$| ( U \cap (V+W+X) )/ (U\cap V + U\cap W + U \cap X) |$$
where I've written |s for "dim". First homology is the direct sum $$( U \cap (V+W+X) )/ (U\cap V + U\cap W + U \cap X) \oplus (V \cap (W+X))/(V \cap W + V \cap X) $$
Example1: 4 planes in R^3, with the intersection of any 3 being {0}. Then 2nd homology is 1-dim and 1st homology is zero. Euler char is 3-8+6=1.
Example2: 4 planes in R^3, three of which meet in a line, the other in "general position". Euler char 3-8+6-1=0, no homology.
This ought to be part of some kind of homology theory for subspace configurations, but it doesn't seem to be in the literature.
Disclaimer: this is all from some ancient notes of mine, and I can't vouch for how reliable it is