Linearity of everything

Question

May I ask for details about how can I prove "linearity of everything" for the following step?

$(f^*dx_i)(Y) = \sum_{j = 1}^lY^j (f^*dx_i)(\frac{\partial}{\partial y^j}) = \sum_{j = 1}^lY^jdx_i(f_*(\frac{\partial}{\partial y^j}))$.

Answer 1

Check out my comments to my answer to this question, clearly the one to which you refer. Feel free to leave comments if you need more info. Cheers, RKL.