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Exercise 7.4.8

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If $ϕ (T)$ is not injective, we can find a nonzero element $x$ such that $ϕ (T) (x) = 0$ . Hence the $T$ -annihilator $p (t)$ divides $ϕ (t)$ by Exercise 7.3.15(b). This means $p (t) = ϕ (t)$ . If $f (t)$ is the characteristic polynomial of $T$ , then we have $f (T) (x) = 0$ by Cayley-Hamilton Theorem. Again by Exercise 7.3.15(b) we have $ϕ (t)$ divides $f (t)$ .

jephianlin

2011-06-27 00:00

Exercise 7.4.8

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