Basis of very ample line bundle

Question

Let $X$ be a projective scheme over field $K$ and $L$ be a very ample line bundle over $X$. Assume that $r=h^0(X,L)$, then can we find $r$ closed points $\{P_1,...,P_r\}$ and a basis $\{s_1,...,s_r\}$ of $H^0(X,L)$ such that $s_i(P_j)=\delta_{i,j}$.