Define T PnR rightarrow Pn1 R by Tf f the derivative transf

Define T: P_n(R) rightarrow P_n-1 (R) by T(f) = f\' (the derivative transformation). (a) Prove that T is a 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 linear

b)

T(f)=0 gives f\'=0

ie N(t)= Set of all constant polynomials

SO basis fo N(T)={1}

T takes polynomial of degree atmost n and differentiates ehnce degree of polynomials in R(T) is atmost n-1

Hence, R(t) = Set of all polynomials of degree n-1

So basis R(t)={1,...,x^{n-1}}