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Let $V$ be a vector space with $\operatorname{dim} V=10$ and let $L: V \rightarrow V$ be a linear transformation. Consider $L^{k}: V \rightarrow V, k=1,2,3, \ldots$ Let $r_{k}=\operatorname{dim}\left(\operatorname{Im} L^{k}\right)$, that is, $r_{k}$ is the dimension of the image of $L^{k}, k=1,2, \ldots$

Give an example of a linear transformation $L: V \rightarrow V$ (or show that there is no such transformation) for which:
a) $\left(r_{1}, r_{2}, \ldots\right)=(10,9, \ldots)$
b) $\left(r_{1}, r_{2}, \ldots\right)=(8,5, \ldots)$;
c) $\left(r_{1}, r_{2}, \ldots\right)=(8,6,4,4, \ldots)$.
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