I have a question about the following answer here: Zero Sectional Curvature implies exp is a local isometry
We have $w\in T_v(T_pM)\cong T_pM$, the geodesics $\gamma_s(t)=\exp_p(t(v+sw))$ and the Jacobi field along $\gamma_0$, $J(t)=(d\exp_p)(tw)$.
If we suppose the sectional curvature is zero then $\dfrac{\partial^2 J}{\partial t^2}=0$.
Let $\{e_i\}$ be an orthonormal basis of $T_pM$ and take the parallel transport along $\gamma_0$, $e_i(t)$ such that $e_i(0)=e_ i$. Then we can write $J(t)=a^i(t)e_i(t)$ and $w=w^ie_i$.
(This is the part I don't understand:) Solving the above equation we have $J(t)=(w^it)e_i(t)$.
How do yo conclude that? Can anyone explain?
Thank you very much.