Let \[f : \mathbb{C}^2 \to \mathbb{C}^2, \quad (z_1,z_2) \mapsto (-z_2,z_1).\] Find the generalised eigenspaces corresponding to the distinct eigenvalues of \(f.\)

Let \(f : V \to V\) be an invertible endomorphism of the \(\mathbb{K}\)-vector space \(V.\) Show that \(\mathcal{E}_f(\lambda)=\mathcal{E}_{f^{-1}}(1/\lambda)\) for all \(\lambda \in \mathbb{K}^*.\)

Let \(V\) be an \(n\)-dimensional \(\mathbb{C}\)-vector space and let \(f:V\to V\) be an endomorphism. Show that if \(f\) is nilpotent, then \(0\) is the only eigenvalue of \(f.\)

Let \(V\) be a \(\mathbb{K}\)-vector space of dimension \(n \in \mathbb{N}\) and \(f : V \to V\) an endomorphism which is not nilpotent. Show that \(V=\operatorname{Ker}(f^{n-1})\oplus \mathrm{Im}(f^{n-1}).\)