Let T:mathbb C^7tomathbb C^7 be a mathbb C-linear operator with eigenvalues 2,3 and 5. Consider the subspace W:=vinmathbb C^7:(T-5I)^kv=0 for some integer k0 of mathbb C^7. Suppose that (T-2I)^2(T-3I)^2(T-5I)^2=0. Which of the following statements are necessarily true?

Q75Section C (Part C)75 of 120

Jordan structure of an operator with a given annihilating polynomial

Let $T:\mathbb C^7\to\mathbb C^7$ be a $\mathbb C$ -linear operator with eigenvalues $2,3$ and $5$ . Consider the subspace

$W:=\{v\in\mathbb C^7:(T-5I)^kv=0\text{ for some integer }k>0\}$

of $\mathbb C^7$ . Suppose that $(T-2I)^2(T-3I)^2(T-5I)^2=0$ . Which of the following statements are necessarily true?

Multiple select: one or more options are correct. Select all of them, then submit — marks need the exact set.

Select every correct option, then submit.

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