## Differential Equations and Linear Algebra (4th Edition)

$T(a+b)=(a+b)''+a_1(a+b)'+a_2(a+b)=a''+b''+a_1a'+a_1b'+a_2a+a_2b=T(a)+T(b)$ $T(ca)=(ca)''+a_1(ca)'+a_2(ca)=c\cdot a''+c\cdot a_1a'+c\cdot a_2a=cT(a)$ Thus it is a linear transformation.