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(adsbygoogle = window.adsbygoogle || []).push({});

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[itex]\in[/itex]M

Proposition: T_{p}F : T_{p}M → T_{F(p)}is linear

ok I know that v[itex]\in[/itex]T_{p}M means that

v:C^{∞}(M)→ℝ is a derivation and that T_{p}M is a vector space.

Does this mean that the image of (av+bw) under T_{p}F where v,w [itex]\in[/itex] T_{p}M and a,b [itex]\in[/itex] ℝ

is aT_{p}F(v) + bT_{p}F(w) which means T_{p}F is linear?

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# Properties of Differentials, Smooth Manifolds.

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