A set of vector \( \left\{\left(a_{1}, a_{2}, a_{3}\right),\left(b_{1}, b_{2}, b_{3}\right),\left(c_{1}, c_{2}, c_{3}\right)\right\} \) is said to be linearly independent if and only if
\[
\left|\begin{array}{lll}
a_{1} & a_{2} & a_{3} \\
b_{1} & b_{2} & b_{3} \\
c_{1} & c_{2} & c_{3}
\end{array}\right| \neq 0,
\]
otherwise the set is said to be linearly dependent.
A similar result holds for \( \left\{\left(a_{1}, a_{2}\right),\left(b_{1}, b_{2}\right)\right\} \).
If \( \left(a_{1}, a_{2}, a_{3}\right),\left(b_{1}, b_{2}, b_{3}\right) \) and \( \left(c_{1}, c_{2}, c_{3}\right) \) are linearly independent and \( x, y, z \in \mathbf{R} \), then
(a) there exist \( \alpha, \beta, \gamma \in \mathbf{R} \) such that \( (x, y, z)+ \) \( \alpha\left(a_{1}, a_{2}, a_{3}\right)+\beta\left(b_{1}, b_{2}, b_{3}\right)+\gamma\left(c_{1}, c_{2}, c_{3}\right) \) \( =(0,0,0) \)
(b) if \( x\left(a_{1}, a_{2}, a_{3}\right)+y\left(b_{1}, b_{2}, b_{3}\right)+z\left(c_{1}, c_{2}, c_{3}\right)= \) \( (0,0,0) \Rightarrow x+y+z \neq 0 \).
(c) \( \alpha\left(a_{1}, a_{2}, a_{3}\right)+\beta\left(b_{1}, b_{2}, b_{3}\right)+\chi\left(c_{1}, c_{2}, c_{3}\right)=(x \), \( y, z) \) cannot hold for any values of \( \alpha, \beta, \gamma \).
(d) none of these.
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