I started studying Riemannian geometry a day ago and in my book it says you form the following bias coordinates in the tangent space.

$\displaystyle e_i = \dfrac{\partial}{\partial x_i}$

And if you have a vector $\displaystyle w(w_1,w_2,\ldots,w_n)$

$\displaystyle w=w_ie_i$

What I don't understand is how can you make an operator in biasing coordinates. Does the above imply that $\displaystyle w=w_ie_i=\dfrac{\partial w_i}{\partial x_i}$ i.e. $\displaystyle v\dfrac{\partial}{\partial x}\equiv\dfrac{\partial v}{\partial x}$