Separating tangent vector

Question

Let $X$ be a projective scheme over an algebraically closed field $k$, $\mathcal{L}$ an invetible sheaf on $X$ and $V \subseteq \mathcal{L}(X)$. I saw in Hartshorne's book(p.152) that elements of $V$ separate tangent vectors if for each closed point $p\in X$, the set $\{s\in V | s_p \in m_p\mathcal{L}_p\}$ spans the $k$ vector space $m_p\mathcal{L}_p/m^2_p\mathcal{L}_p$. But, I don't know why do this fact separate tangent vectors..

Answer 1

In other words, for any two distinct tangent vectors (i.e. elements of $m_p L_p / m_p^2 L_p$ ) we can find two different sections of the line bundle which vanish at the point p (i.e. they are in the maximal ideal at $p$) whose reduction mod $m_p ^2 L $ is the tangent vector.