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QUESTION Show that the only solution of the equation
$$\lambda(0,2,3)+\mu(4,5,6)+\nu(7,8,9)=(0,0,0)$$ is
$\lambda=\mu=\nu=0$



ANSWER This is equivalent to solving

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

The coefficient matrix $A$ has non-zero determinant, so it can be
inverted and the solution is $A^{-1}\textbf{0}=\textbf{0}$.



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