Fun with linear algebra
Epistemic status: Mostly the result of a late night procrastination session. Proceed with caution. Might be too verbose or not verbose enough.
I have been thinking about linear algebra quite a lot lately. There is a common proof strategy, which usually goes something like:
- You have a subspace of a vector space
- Get a basis for that subspace
- Use the exchange lemma to extend that basis to a basis of some larger subspace
- Try to figure out if your new basis, or maybe the basis of the complementary subspace, has some desirable properties
However, after reading a particularly cool proof (3.106) in Linear Algebra Done Right, I noticed a fascinating concept: many linear algebra theorems, which are commonly proven using the startegy described above, can often be proven by picking a bunch of appropriate linear maps between the relevant vector spaces and then using a bunch of “elementary” theorems, usually proven using the above method.
This usually takes a bit of effort to figure out, but the results are quite satisfying. Let me illustrate:
Proposition (Dimension of a sum): Let and be finitely generated subspaces of a common vector space. Then we have
Proof. Consider a pair of linear maps:
It is clear that is surjective, and it is also clear that is injective. We just need to prove that .
Take . Then we compute therefore .
Now we consider . This means that Additionally, is a subspace, hence it contains the additive inverse for each of its elements, meaning . This means we can use to get and therefore .
We finish by using the Rank-Nullity theorem twice, first on and then on
The idea is ultimately simple: we know that every element in can be expressed as a sum of a vector from and a vector from . By investigating the kernel of , we are essentially asking how many ways this can be done.
I am not really sure if constructing proofs like this one is really “practical”. However, I find this approach somewhat more aesthetic and enlightening than picking a bunch of basis and then getting lost in a sea of coordinates.
Anyway, this blog post is pretty different from what I used to write in the past. I have finished my Electrical Engineering bachelor’s degree and started working towards getting a bachelor’s degree in Mathematics, so I ultimately hope to post more in this vein in the future. I do hope to write some technical posts as well, I have not given up engineering completely and still continue to work as a freelance embedded systems engineer.