## Exact sequences split

Now that we know the splitting lemma, we can show that *every* short exact sequence of vector spaces splits!

To see this, we’ll need to refine an earlier result. Remember how we showed that every vector space has a basis. We looked for *maximal* linearly independent sets and used Zorn’s lemma to assert that they existed.

Here’s how we’re going to refine this result: start with a collection of linearly independent vectors. Then we don’t just look for any maximal collection, but specifically for a maximal collection containing . Clearly if we have a linearly independent set containing which is maximal among such sets, it is also maximal among all linearly independent sets — it is a basis. On the other hand, the previous argument (with Zorn’s lemma) says that such a maximal linearly independent set must exist.

What does this mean? It says that *any linearly independent set can be completed to a basis*. If we start with the empty set (which is trivially linearly independent) then we get a basis, just like before. So we recover the same old result as before.

But look what we can do now! Take a short exact sequence

and pick any basis of (notice that we’re using a generic, possibly infinite, index set). Now hit this basis with to get with . I say that this is a linearly independent set in .

Why is this? Well, let’s say that there’s a linear combination (only finitely many of the can be nonzero here). This linear combination is in the image of , since we can write it as . But exactness tells us that is injective, and so we have . But then all the have to vanish, since the form a basis!

So we’ve got a linearly independent set in . We now complete this to a maximal linearly independent set with . This is now a basis for , which contains the image of all the basis elements from .

Now turn this around and define a linear transformation by specifying its values on this basis. Set for and for . Then the composition is the identity on (check it on our basis), and so the sequence splits, as we said.

Notice here that all the Zorniness only matters for infinite-dimensional vector spaces. Everything we’ve done here works in without ever having to worry about such set-theoretic problems. However, given my politics I have no problem with using the Axiom of Choice when push comes to shove. I’m just pointing this out in case you’re the squeamish sort.