Properties of Differentials, Smooth Manifolds.

[BrainHurts]

I'm reading the second edition of John M. Lee's Introduction to Smooth Manifolds and he has a proposition that I'd like to understand better

Let M, N, and P be smooth manifolds with or without boundary, let F:M→N and G:N→P be smooth maps and let p$\in$M

Proposition: T_pF : T_pM → T_F(p) is linear

ok I know that v$\in$T_pM means that

v:C^∞(M)→ℝ is a derivation and that T_pM is a vector space.

Does this mean that the image of (av+bw) under T_pF where v,w $\in$ T_pM and a,b $\in$ ℝ

is aT_pF(v) + bT_pF(w) which means T_pF is linear?

 

[micromass]

Yes, that's what it means.