VI.6) (*) Determine if the following are true or false and think of a brief explanation of why
that is the case.
(i) Given $S, T \in \mathcal{L}(V)$ two nilpotent operators, then $ST$ is nilpotent.
(ii) Let $T \in \mathcal{L}(V)$ be nilpotent and diagonalizable, then $T = 0$.
(iii) Given $S, T \in \mathcal{L}(V)$ two nilpotent operators, then $S + T$ is nilpotent.
(iv) Given $S, T \in \mathcal{L}(V)$ two nilpotent operators such that $ST = TS$, then $S + T$ and $ST$
are nilpotent.
(v) Let $T \in \mathcal{L}(V)$, assume that there exists $B_V$ a basis of $V$ such that $\mathcal{M}(T, B_V)$ is a
diagonal matrix. Then for any Jordan basis $B_V'$ the matrix $\mathcal{M}(T, B_V')$ is diagonal.
(vi) Let $T \in \mathcal{L}(V)$ on a complex vector space and consider $B_V$ and $B_V'$ two different
Jordan basis. Then $\mathcal{M}(T, B_V)$ and $\mathcal{M}(T, B_V')$ can have a different number of blocks
in its diagonal form.
