Suppose $\wp_n(\Re)$ is the space of polynomials with degree $n$. Let $T:\wp_n(\Re) \rightarrow \wp_n(\Re)$ being $T(p(x)) = xp'(x)$.
How can I prove that T is a linear transformation?
I know I should prove that:
- $T(u_1 + u_2) = T(u_1) + T(u_2)$, $\forall u_1, u_2 \in \wp_n(\Re)$
- $T(\lambda u) = \lambda T(u) $, $\forall \lambda \in \Re$ and $\forall u \in \wp_n(\Re)$
However, since $T(p(x))$ is a function with $p(x)$ as the argument, I'm not sure how to deal with de $x$ on $xp'(x)$.