Opgave 1c. Recall the representation of \(k[X]\) as in Exercise 8(c) of Week 36. Explain that any finite dimensional representation \(V\) of \(k[X]\) is of this form.
Bewijs. Let \(V\) be a finite dimensional representation of \(k[X]\). Then there exists a ringhomomorphism \(\rho: k[X] \to \operatorname{End}_k(V)\), we are going to show that \(X.v = Av\) for some \(A \in \operatorname{End}_k(V)\) for all \(v\in V\). But this follows directly as \(\rho\) maps \(X\) to an element \(A\in \operatorname{End}_k(V)\).