Find a basis px qx for the kernel of the linear transformati

Find a basis [p(x) q(x)] for the kernel of the linear transformation L:epsilon P_3 [x] R| defined by L(f(x)) =f\'(2) -f(1)| where P_3[x] is the vector space of polynomials in x with degree less than 3.

Solution

L(a+bx+cx^2+dx^3)=(b+2cx+3dx^2)(2)-(a+b+c+d)=-a-c-d+4c+12d=-a+3c+11d=0

a=3c+11d

f(x)=3c+11d+bx+cx^2+dx^3=c(3+x^2)+bx+d(11+x^3)

Basis is

{3+x^2,x,11+x^3}