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I have this list $$ S = ((1,2,0,1), (0,1,0,0), (1,0,0,2))$$ and I need to prove it doesn't generate whole vector space on $\mathbb R^4$. (I know how to prove generating for 4 vectors.) But what example I have to use?

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Without using general theorems, you can see every linear combination of these vectors has a third coordinate of $0$. Therefore they cannot generate the whole space because there exist vectors with third coordinate nonzero.

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You also can prove this without using any linear combination between the vectors of your list. You simply can't generate $\mathbb R^4$ using a list of only three vectors (you will need at least four of them), because $\dim(\mathbb R^4)=4$.

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