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QUESTION Find non-zero $\lambda,\mu, \nu$ such that
$$\lambda(1,2,3)+\mu(4,5,6)+\nu(7,8,9)=(0,0,0)$$


ANSWER This is equivalent to solving

\begin{eqnarray*}
\lambda+4\mu+7\nu&=&0\\ 2\lambda+5\mu+8\nu&=&0\\
3\lambda+6\mu+9\nu&=&0\\
\end{eqnarray*}

which has $(\lambda, \mu, \nu)=(1, -2, 1)$ as a solution (or any
multiple).



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