Give an example of a linear transformation \(L: V \rightarrow V\) (or show that there is no such transformation) for which:

Question

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) \(\left(r_{1}, r_{2}, \ldots\right)=(8,6,4,4, \ldots)\).