in my book tangent vectors are defined as equivalence classes of curves that are tangent to the given vector. let σ be a curve on a manifold and h*([σ]) is defined to be [h o σ]. v1 = [σ1] and v2 = [σ2]. show that the pushforward is linear: h*(v1 + v2) = h*(v1) + h*(v2) and h*(rv1) = rh*(v1).

i am starting this problem with showing the first part h*(v1 + v2) = h*(v1) + h*(v2) and i am having trouble. earlier my book defines v1 + v2 = [φ^-1 (φ o σ1 + φ o σ2)] where φ is a coordinate function.

i have that h*(v1+v2) = [h(φ^-1 (φ o σ1 + φ o σ2))] but i don't know how to get from there to h*(v1) + h*(v2) = [h o σ1] + [h o σ2]. if φ^-1 was linear then it would work but that doesn't seem to be the case all the time however.