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?
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.
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 $\dim(\mathbb R^4)=4$.