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I am looking for a collection of $5$ vectors in $\mathbb{R}^3$ such that any $3$ are linearly independent. What I tried to do at first is set up a collection of linear equations like this: $$c_1M_1 + c_2M_2 + c_3M_3 \ne 0$$ $$c_1M_1 + c_2M_2 + c_4M_4 \ne 0$$ $$c_1M_1 + c_2M_2 + c_5M_5 \ne 0$$ $$c_2M_2 + c_3M_3 + c_4M_4 \ne 0$$ $$\ldots$$

and so on for any combination, which is retrospectively obviously incorrect. So I think I need $5$ vectors, each with $3$ elements that are all linearly independent. But this doesn't seem possible to me. Can anyone shed light on what I am supposed to do?

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Try

$$(1,0,0)$$ $$(0,1,0)$$ $$(0,0,1)$$ $$(1,1,1)$$ $$(1,2,3)$$

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