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Exercise 8.A.6

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Proof. Suppose by contradiction that $S \in L (ℂ^{3})$ is a square root of $T$ . Note that $V = null T^{3}$ . We have

\begin{aligned} V & = null T^{3} \\ = null R^{6} \\ = null R^{3} \\ = null RT \\ \subset null R^{2} T \\ = null T^{2}, \end{aligned}

where the third line follows by 8.4. But this a contradiction, since $T^{2} (z_{1}, z_{2}, z_{3}) = (z_{3}, 0, 0)$ , we see that $null T^{2} = {(0, 0, z) : z \in ℂ}$ , then we can’t have $V \subset null T^{2}$ . □

celiopassos

2017-10-06 00:00

Exercise 8.A.6

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