Exercise 10, § 7

Let E be a vector space over a field K. Show that every endomorphism u of E is an automorphism or can be written as $u = v - w$, where v and w are automorphisms of E. (Observe that u can always be replaced by $s_1 \circ u \circ s_2$, where $s_1$ and $s_2$ are two automorphisms of E. Distinguish two cases, according to whether $\dim(\mathrm{Ker}(u)) \leq \dim(\mathrm{Coker}(u))$ or $\dim(\mathrm{Ker}(u)) > \dim(\mathrm{Coker}(u))$; when $\mathrm{rg}(u)$ is finite, then the first case always holds. When

it can be reduced to the case where $\mathrm{Ker}(u) \cap \mathrm{Im}(u) = \{0\}$; if it can also be assumed that $\mathrm{Im}(u) + \mathrm{Ker}(u) = E$ and then apply Exercise 9 (c). If $\dim(\mathrm{Ker}(u)) < \dim(\mathrm{Coker}(u))$, there is a supplementary subspace in E of $\mathrm{Ker}(u)$ of the form $W \oplus \mathrm{Im}(u)$ with $\dim(W) = \dim(\mathrm{Im}(u)) = \dim(E)$; apply the remark at the beginning with $s_1(u(\mathrm{Im}(u))) = \mathrm{Im}(u)$ and use Exercise 9 (a) and (c). When $\dim(\mathrm{Ker}(u)) > \dim(\mathrm{Coker}(u))$, it can be reduced to the case where $\mathrm{Ker}(u) \subset \mathrm{Im}(u)$; this time take $s_1(\mathrm{Ker}(u)) = u^{-1}(\mathrm{Ker}(u))$ and use the results of the preceding cases and Exercise 9.)