If I have an endomorphism between vector spaces $f:V \rightarrow V$, such that $Im(f)+Ker(f) \neq dim(V)$, is this equivalent to $f$ being nilpotent?
2026-03-25 13:13:29.1774444409
nilpotent endomorphism and $Im(f)+Ker(f) \neq dim(V)$
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No, it isn't. Check out $$ A = \begin{pmatrix}0 & 1 & 0\\0 & 0 & 0\\0 & 0 & 1\end{pmatrix}. $$