How do you prove that $P(X) \mapsto P(X+1) - P(X)$ is a linear map?

Question

I know what I have to do to show that the map is linear, but I'm just not sure how to handle the operations with polynomials.

Do I just have to take two polynomials, say, $P(X)$ and $Q(X)$ and describe them as $ P(X) = a_0 +a_1X + ... + a_nX^n$ and $Q(X) = b_0 + b_1X + ... + b_mX^m$ and prove it that way, or is there a simpler method? Thanks.

Answer 1

There is a simpler method. Let $\Psi$ be your map. If $P(x)$ and $Q(X)$ are polynomials, then\begin{align}\Psi\bigl(P(X)+Q(X)\bigr)&=P(X+1)+Q(X+1)-P(X)-Q(X)\\&=P(X+1)-P(X)+Q(X+1)-Q(X)\\&=\Psi\bigl(P(X)\bigr)+\Psi\bigl(Q(X)\bigr).\end{align}Can you do the rest?

Answer 2

$P(X) \mapsto P(X)$ is a linear map(the identity), and so is $P(X) \mapsto P(X+1)$, because $(P+Q)(X+1) = P(X+1) + Q(X+1)$, and $(cP)(X+1) = c \cdot P(X+1)$. And, the sum or difference of two linear functions is linear.