- #1
juanma101285
- 5
- 0
Hi, I have the following problem that is solved, but I get lost at one step and cannot find how to do it in the notes. I would really appreciate it if someone could tell me where my teacher gets the result from.
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!
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!