The name "undergraduate linear algebra" in the title is a bit of a joke, and so I don't know how widely spread it is. To wit:

**High school linear algebra**is the theory of a finite-dimensional vector space — i.e. it consists of the finite-dimensional representation theory of a field $\mathbb F$, or, better, for representation theory over $\mathbb F$ of the (trivial) monoid of size $1$ (the free monoid on zero generators). The complete structure theory is well-known: every finite-dimensional vector space is (noncanonically isomorphic to) a direct sum of copies of the one-dimensional (free) vector space: $V \cong \mathbb F^n$ for some integer $n$. In fact, assuming the axiom of choice, one can extend this result to arbitrary vector spaces: if $\mathbb F$ is a field and $V$ is a $\mathbb F$-module, then $V \cong \mathbb F^\kappa = \coprod^\kappa \mathbb F$ for some cardinal $\kappa$.**Undergraduate linear algebra**is the theory of a finite-dimensional vector space with a choice of matrix, which is to say it is the finite-dimensional representation theory of the polynomial ring $\mathbb F[x]$, or equivalently the finite-dimensional representation theory of the free monoid $\mathbb N$ on one generator. When $\mathbb F$ is algebraically closed, the complete structure theory is given by the Jordan Canonical Form theorem. The category of finite-dimensional representations of $\mathbb N$ is not semisimple, but the indecomposables are parameterized by pairs $(\lambda,n)$ where $\lambda \in \mathbb F$ and $n\in \mathbb Z_{>0}$ representing "Jordan blocks", and the irreducibles are precisely the indecomposables with $n = 1$, so that the irreducibles are parameterized by $\lambda \in \mathbb F$. If $\lambda \neq \mu$, then the irreducibles corresponding to $\lambda,\mu$ have no nontrivial extensions; there is a unique nontrivial extension of the irreducible corresponding to $\lambda$ by itself.

I recently realized, however, that I never learned much of anything about the infinite-dimensional representation theory of the monoid $\mathbb N$, even assuming that my field $\mathbb F$ is algebraically closed (and, if you like, characteristic zero), and even assuming the axiom of choice. There are plenty of vector spaces with endomorphisms that have no eigenvectors — a standard example with $\mathbb F = \mathbb C$ is $V = $ the vector space of smooth $\mathbb C$-valued functions on $\mathbb R$ that vanish outside the interval $[0,1]$, and the endormorphism is differentiation.

With some restrictions (words like "compact operator" and "spectral theorem" come up), there are lots of results that try to reproduce the Jordan theorem. But since I hate all types of analysis, my question is:

Assuming the axiom of choice, to what extent is it possible to describe the full (infinite-dimensional) $\mathbb C$-representation theory of the monoid $\mathbb N$? (Equivalently, the category of $\mathbb C[x]$-modules.)

A good answer might consist of a minimal "generating" set, i.e. a set $\mathcal S$ of $\mathbb N$-representations to that every $\mathbb N$-rep is a colimit of a diagram whose objects are all in $\mathcal S$. (E.g. for infinite-dimensional high school algebra, $\mathbb S$ consists of the one-dimensional vector space.) I could also imagine an answer consisting of a "no-go theorem" that the representation theory of $\mathbb N$ is hard; maybe that's why I never learned it.

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