Define T PnR rightarrow Pn 1 R by Tf the derivative transfo

Define T: P_n(R) rightarrow P_n - 1 (R) by T(f)\' (the derivative transformation). (a) Prove that T is linear transformation. You can use what you know from calculus without reproving it. (b) Find bases for N(T) and R(T).

Solution

a)

T(f+g)=(f+g)\'=f\'+g\'=T(f)+T(g)

T(cf)=(cf)\'=cf\'=cT(f)

Hence, T is a linear transformation

b)

T(f)=0

f\'=0

f=C ie constant polynomials

N(t)=Set of all constant polynomials

So basis for N(T)={1}

R(T) =Set of all polynomials of degree n-1

So basis for R(T)=Standard basis for R_{n-1}={1,...,x^{n-1}}