# A linear subspace of projective space

If we have a linear subspace of projective $P^n_k$ where $k=\bar k$ that contains all the points $[0:\ldots:0:x:0:\ldots:0]$ with the nonzero $x$ in the $i$th slot for all $i$, how can we see that this is actually the whole space.

E.g. in $P^2$, suppose $L$ is a linear subspace which contains the points $[x:0:0]$ and $[0:x:0]$ and $[0:0:x]$. How do we see that $L$ is in fact equal to $P^2$.

Thanks!

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Replacing everywhere $x$ by $1$ doesn't change the problem but makes the question clearer by eliminating an irrelevant letter. Also, $k=\bar k$ is irrelevant. –  Georges Elencwajg Mar 29 '13 at 13:01

No hyperplane $l(z)=\sum c_jz^j=0$ contains all your points $P_i$: substitution yields $l(P_i)=c_i=0$ .