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The problem says:

"Find the matrix of linear mapping [itex]T:P_3 → P_3[/itex] defined by

[itex](Tp)(t)=p(t)+p'(t)+p(0)[/itex]

with respect to the basis {[itex]1,t,t^2,t^3[/itex]} of [itex]P_3[/itex]. Deduce that, given [itex]q \in P_3[/itex], there exists [itex]p \in P_3[/itex] such that

[itex]q(t)=p(t)+p'(t)+p(0)[/itex]."

And I get lost here... It says:

"We have

[itex]T(1)=2[/itex]

[itex]T(t)=1+t[/itex]

[itex]T(t^2)=2t+t^2[/itex]

[itex]T(t^3)=3t^2+t^3[/itex]"

So I don't know why it says [itex]T(1)=2[/itex].... I think [itex]T(t)=1+t[/itex] because it is the derivative of t plus t, and [itex]T(t^2)[/itex] is the derivative of [itex]t^2[/itex] plus [itex]t^2[/itex]... But why T(1)=2?

Thanks a lot!