The endomorphism \(f=f_{\mathbf{J}_n(\lambda)}: \mathbb{C}^n\to\mathbb{C}^n\) is nilpotent if \(\lambda = 0.\)

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If \(f=f_{\mathbf{J}_n(0)}: \mathbb{C}^n\to\mathbb{C}^n, n>1\) it holds that \(\dim(\operatorname{Ker}(f))>1.\)

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• False

If \(f:\mathbb{C}^n\to\mathbb{C}^n\) is nilpotent, then \(f=f_{\mathbf{J}_n(0)}\) or \(f=o.\)

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• False

The endomorphism \(\frac{\mathrm d}{\mathrm dx}:\mathsf P_n(\mathbb{R})\to\mathsf P_n(\mathbb{R})\) is nilpotent.

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If \(V\) is a complex vector space and \(f:V\to V\) is a nilpotent endomorphism, then \(\operatorname{Tr}(f) = 0.\)

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• False

If \(V\) is a complex vector space and \(f:V\to V\) is a nilpotent endomorphism, then \(f\) is diagonalisable if and only if \(f=o.\)

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• False

If \(V\) is a complex vector space, there exists a nilpotent endomorphism \(f:V\to V\) such that \(f(v) = v\) for some \(v\ne 0_V.\)

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• False

If \(m\in \mathbb{N}\) is the smallest number such that \(\mathbf{A}\in M_{n,n}(\mathbb{R})\) satisfies \(\mathbf{A}^m=\mathbf 0_n,\) then \(m\leqslant n.\)

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Let \(f:V\to V\) be a nilpotent endomorphism. Then \(1\) is the only eigenvalue of \(f+\mathrm{Id}_V.\)

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• False

Let \(f:V\to V\) be an endomorphism. If \(\dim(\operatorname{Im}(f))<\dim(V),\) then \(f\) must be nilpotent.

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• False