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.

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