Thread: Need help proving that differentiation is a linear transformation

Define D ∈ L(p(R), p(R)) by (Dp)(r)=p'(r) for all r∈R and for all p∈p(R).

I know that to show that something is a linear transformation you need to show that F(v1+v2)=F(v1)+F(v2) and also F(av)=aF(v). However I do not understand HOW to show this. Can someone help with this please? I just am not sure how exactly to get started. It would be great if perhaps someone could do the first few lines or so, I think I would be able to finish if I could only get a start.

Here is my attempt at the first few lines...can someone just let me know if it is correct or not? (It's probably not lol)

(D(v1+v2))(r)=(v1+v2)'(r)
=(v1(r)+v2(r))'
=(v1(r))'+(v2(r))'

Hopefully you learned in your first Calculus class that (f+ g)'= f+ g' (what you are saying in your second post) and that (cf)'= cf' for c a constant. That's exactly what you need.