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## Main Question or Discussion Point

Say I have to vector spaces [itex]V,W[/itex] and a linear transformation [itex]\Phi:V\rightarrow W[/itex]. I know that (given [itex]v,p\in V[/itex]) if I interpret a tangent vector [itex]v_p[/itex] as the initial velocity of the curve [itex]\alpha(t)=p+tv[/itex] I have, relative to a linear coordinates system on [itex]V[/itex], [itex]v_p=x^i(v)\partial_{i(p)}[/itex].

The thing I don't understand is why in this case [itex]d\Phi (v_p)=(\Phi(v))_{(\Phi(p))}[/itex]. Can someone show me the way?

The thing I don't understand is why in this case [itex]d\Phi (v_p)=(\Phi(v))_{(\Phi(p))}[/itex]. Can someone show me the way?