Is $T(P(x))=P'(X)+(x-2)P(X)$ a linear transformation? Could anyone help me with this question?
Let $P_2[\mathbb R],P_3[\mathbb R]$ be the spaces of polinomyals $a_0+a_1x+...+a_nx^n$ where $n \leq 2,n \leq3$ respectively and let  $T:P_2[\mathbb R] \rightarrow P_3[\mathbb R]$ be a transformation defined by: 

$T(P(x))=P'(X)+(x-2)P(X)$.

Is $T$ a linear transformation? If it is, prove it, if not, show a counterexample.
Thank you!

Let $P,Q$ polynomials and $a,b$ reals.
$
T(aP + bQ)  
$ is defined as
$$T(aP + bQ)(X)
= (aP + bQ)'(X)+(X-2)(aP + bQ)(X)\\
= aP'(X) + bQ'(X) + (X-2)aP(X) + (X-2)bQ(X)\\
T(aP + bQ)(X)= aT(P)(X) + bT(P)(X)
$$hence $T(aP + bQ)=aT(P) + bT(P)$: $T$ is linear.
