Fun with polynomials and linear algebra; or, slight abstract nonsense

Posted 2026-04-30

This is mostly a bunch of notes to myself (with some slight expansion) and is a combination/extension/simplification of theorems/ideas/constructions from a bunch of texts, including Wistbauer’s “Foundations of Module and Ring Theory” and Fuhrmann’s “A Polynomial Approach to Linear Algebra”, along with others that at this point I don’t recall.

While most of the things here are pretty standard (indeed, many of these are just slightly generalized definitions given in introductions to linear algebra over arbitrary fields!) there is some stuff here that might be weird and surprising unless you’re already quite well-versed in the language of module theory. (We won’t be discussing modules here, though, even if many statements have natural module-theoretic generalizations.)

This document/post is more of an exercise to see just how many “standard” constructions can be done purely in the linear-algebraic language. In a sense, this document is an “any% speedrun” of some theorems (Chinese remainder theorem, e.g.) and tidbits that most of us associate with different parts of math. It will also have some small exercises littered throughout, which are fun one-or-two-liners for those interested.

Anyways, onwards.

Basic facts

Let V be a vector space over a field 𝐅 in what follows. If V ′ is a vector space over the same field, we say

V ≃ V ′

if there exists any invertible linear map between V and V ′ . It is pretty easy to see that this is an equivalence relation.

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