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Is there an example of a $3\times 3$ matrix $M = [v_{1}\, v_{2}\, v_{3}]$ (where $v_{i}$ are $3 \times 1$ vectors) such that $v_{1}, v_{2}, v_{3}$ are linearly dependent however if we have $av_{i} + bv_{j} = 0$ then $a, b = 0$ for each $(i, j)$ pair?

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Yes: $$v_1 = (1,0,0)$$ $$v_2 = (1,1,0)$$ $$v_2 = (0,1,0)$$

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Yes, $v_1=[1,2,3]$, $v_2=[3,2,1]$ and $v_3=[4,4,4]$. Clearly the three vectors are dependent. However it is trivial to check that pairwise they are independent.

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In general, any three vectors that span a two dimensional space, but such that there is no vector a multiple of another.

For a subset of two such vectors will still span a two dimensional space, hence be linearly independent.

Both other answers fall in this category.

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