Exercise 7.3.9

Use Theorem 7.13. We know that $V$ is a $T$-cyclic subspace if and only if the minimal polynomial $p(t)={(-1)}^{n}f(t)$, where $n$ is the dimension of $V$ and $f$ is the characteristic polynomial of $T$. Assume the characteristic polynomial $f(t)$ is

where $n_i$ is the dimension of the eigenspace of ${\lambda }_i$ since $T$ is diagonalizable. Then the minimal polynomial must be

So $V$ is a $T$-cyclic subspace if and only if $n_i=1$ for all $i$.