Let \(f,g:V\to V\) be endomorphisms of \(V.\) Show that if \(f\circ g=g\circ f,\) then \(\operatorname{Im}(g)\) and \(\operatorname{Ker}(g)\) are stable under \(f.\)

Let \(V\) be a \(\mathbb{K}\)-vector space and \(f\in\operatorname{End}_{\mathbb{K}}(V)\) such that \(f\circ f = f.\) Show that \(V = \operatorname{Im}(f) \oplus \operatorname{Ker}(f).\)

Give a proof of Proposition 6.8.

An endomorphism \(f : V \to V\) of a \(\mathbb{K}\)-vector space \(V\) is called nilpotent if \[f^k=\underbrace{f \circ f \circ \cdots \circ f}_{k\text{ times}}=o_V\] for some \(k \in \mathbb{N}\) and where \(o_V\) denotes the zero endomorphism of \(V.\) Assume \(f : V \to V\) is nilpotent and let \(U^i=\operatorname{Im}(f^i).\)

1. Show that if \(U^i\neq \{0_V\},\) then \(\dim U^{i+1}<\dim U^i\)

2. Suppose \(\dim V=n.\) Show that \(f^n=o_V.\)