设 $\boldsymbol{A}=\left(\begin{array}{rrr}1 & -1 & -1 \\ -1 & 1 & 1 \\ 0 & -4 & -2\end{array}\right), \boldsymbol{\xi}_{1}=\left(\begin{array}{r}-1 \\ 1 \\ -2\end{array}\right)$.
( I ) 求满足 $\boldsymbol{A} \boldsymbol{\xi}_{2}=\boldsymbol{\xi}_{1}, \boldsymbol{A}^{2} \boldsymbol{\xi}_{3}=\boldsymbol{\xi}_{1}$ 的所有向量 $\boldsymbol{\xi}_{2}, \boldsymbol{\xi}_{3}$;
( II ) 对 ( I ) 中的任意向量 $\boldsymbol{\xi}_{2}, \boldsymbol{\xi}_{3}$, 证明 $\boldsymbol{\xi}_{1}, \boldsymbol{\xi}_{2}, \boldsymbol{\xi}_{3}$ 线性无关.
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