One to one correspondence between all $F[x]$-module $V$ and all linear transformations $T\colon V\to V$, $V$ being a vector space over $F$.

If $V$ is a $k[x]$-module, then multiplication with $x$ defines a linear transformation $T: V \to V, v \mapsto x v$.

Conversely, if $V$ is a $k$-vector space and $T:V \to V$ is a linear transformation, then you can define a $k[x]$-module structure by letting a polynomial $p \in k[x]$ act on $V$ by $$ p \cdot v = p(T)(v) $$

The point is that no matter what you start with (a $k[x]$-module structure or a linear transformation) you always get the other thing for free.

Tags:

Linear Algebra

Modules

Abstract Algebra

Linear Transformations

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