# Differential of a Linear Map

Let $V$ be a vector space with linear isomorphisms to $\mathbb{R}^n$ as charts. Let $\alpha (t) = p + tv$ be a curve $\alpha : \mathbb{R} \longrightarrow V$.

Then

$v_p = \alpha\prime(0) \in T_p(V)$ and $d\phi(\alpha\prime(0)) = (\phi \circ \alpha)\prime (0)$.

But $\phi$ is linear (see the exercise) and so

$(\phi \circ \alpha)'(0) = (\phi (p) + t\phi (v))\prime (0) = (\phi (v))_{\phi (p)}$

as was required to be shown.