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Find $3$ vectors $a$, $b$ and $c$ in $\mathbb{R}^3$ such that {$a$, $b$}, {$a$, $c$} and {$b$, $c$} are each linearly independent sets of vectors, but the set {$a$, $b$, $c$} is linearly dependent.

Is there a better way of finding a solution to this problem than just "guessing and check"?

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    $\begingroup$ Look for easy dependences. Suppose any two of your vectors spans the usual $xy$ plane. $\endgroup$ – lulu Apr 22 '16 at 11:38
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How about taking $a,b$ to be any two linearly independent vectors, and then let $c=a+b$.

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