Proving the tangent map is linear Let $M$ and $N$ be some vector spaces and define the map $f:M \rightarrow N$, which is a smooth diffeomorphism. Let $Tf$ be the induced mapping between the tangent spaces of $M$ and $N$. Then it is said that the map $$Tf: T_pM \rightarrow T_{f(p)}N$$ is a linear mapping. This is where the confusion comes in, it may be silly but i don't see how this is a linear map, i know i have to check the conditions for linearlity but this looks like a abstract object, so i'm not sure how to apply the conditions to check the linearity of the map $Tf$. 
Thanks !

Yes, this is an abstract object but it has a specific definition. Apply Taylor's theorem $$f(x+\lambda y)=f(x)+\frac{\partial f}{\partial x}\cdot (\lambda y)+\mathcal{o}\left(|\lambda|\right)$$ So your object $$Tf:=\frac{\partial f}{\partial x}$$ Do you see why this is a linear operator?
