Opgave 8c. Let \(V\) be a finite dimensional \(k\)-vector space and let \(A \in \operatorname{End}_k(V)\). Show that \(V\) admits a unique \(k[X]\)-module structure such that \(X.v = A(v)\) for all \(v\in V\)
Bewijs.

So we need to show there is only way to define \(f.v\) if \(X.v=Av\) for all \(v \in V\). But because of the properties of the action (induced by the ring homomorphism) and because of the linearity of this action, we get:

\[\begin{aligned} f.v &= (\sum_{i=0}^n a_i X^i).v \\ &= \sum_{i=0}^n (a_i X^i).v \\ &= \sum_{i=0}^n a_i (X^i).v \\ &= \sum_{i=0}^n a_i A^iv\end{aligned}\]

Note that the following are true:

\[\begin{aligned} (a+b).v = a.v + b.v \\ (\alpha a).v = \alpha (a.v) \\ (X^i).v = X^{i-1}(X.v) \\\end{aligned}\]