In combinatorics we have a method of determining the cardinality of unions in terms of the sets in question and their intersection: the Inclusion-Exclusion Principle. Predictably enough, this formula is reflected in the subspaces of a vector space. We could argue directly in terms of bases, but it’s much more interesting to use the linear algebra we have at hand.
Let’s start with just two subspaces and of some larger vector space. We’ll never really need that space, so we don’t need to give it a name. The thing to remember is that and might have a nontrivial intersection — their sum may not be direct.
Now we consider the sum of the subspaces. Every vector in is the sum of a vector in and a vector in . This tells us that there’s a linear map defined by . Note carefully that this is linear as a function of the pair . It’s not bilinear — linear in each of and separately — which would mean bringing in the tensor product.
This function is always surjective, but it may fail to be injective. Specifically, if and have a nontrivial intersection then it gives us a nontrivial kernel. If then . But rather than talk about elements, let’s use the intersection to parametrize the kernel! We’ve got another linear map , defined by .
Now it’s clear that has a trivial kernel. On the other hand, its image is precisely the set of pairs that kills off. That means we’ve got an exact sequence:
Taking the Euler characteristic we find that
Some simple juggling turns this into
which is the inclusion-exclusion principle for two subspaces.
What about three subspaces , , and ? We start off again with
defined by . Then the kernel contains triples like , , and . We use the pairwise intersections to cover the kernel by the map
defined by . The image of is exactly the kernel of , but this time it has a nontrivial kernel itself! Now we need another map
defined by . Now is injective, and its image is exactly the kernel of . We stick these maps all together to have the long exact sequence
Taking the Euler characteristic and juggling again we find
We include the dimensions of the individual subspaces, exclude the dimensions of the pairwise intersections, and include back in the dimension of the triple intersection.
For larger numbers of subspaces we can construct longer and longer exact sequences. The th term consists of the direct sum of the intersections of the subspaces, taken at a time. The th map consists of including each -fold intersection into each of the -fold intersections it lies in, with positive and negative signs judiciously sprinkled throughout to make the sequence exact.
The result is that the dimension of a sum of subspaces is the alternating sum of the dimensions of the -fold intersections. First add the dimension of each subspace, then subtract the dimension of each pairwise intersection, then add back in the dimension of each triple intersection, and so on.
From here it’s the work of a moment to derive the combinatorial inclusion-exclusion principle. Given a collection of sets, just construct the free vector space on each of them. These free vector spaces will have nontrivial intersections corresponding to the nonempty intersections of the sets, and we’ve got canonical basis elements floating around to work with. Then all these dimensions we’re talking about are cardinalities of various subsets of the sets we started with, and the combinatorial inclusion-exclusion principle follows. Of course, as I said before we could have proved it from the other side, but that would require a lot of messy hands-on work with bases and such.
The upshot is that the combinatorial inclusion-exclusion principle is equivalent to the statement that exact sequences of vector spaces have trivial Euler characteristic! This little formula that we teach in primary or secondary school turns out to be intimately connected with one of the fundamental tools of homological algebra and the abstract approach to linear algebra. Neat!