Show that the function $\displaystyle T : \mathbb{P}_3(\mathbb{R})\rightarrow \mathbb{P}_3(\mathbb{R})$ defined by

$\displaystyle T(p)=4p'+3p$ where $\displaystyle p'(x)=\frac{dp}{dx}$

is a linear map

Would I begin the question like the following:

$\displaystyle p(x)=a_0+a_1x+a_2x^2+a_3x^3\Rightarrow p'(x)=a_1+2a_2x+3a_3x^2$

$\displaystyle T\begin{pmatrix}a_0\\a_1\\a_2\\a_3\end{pmatrix}=4\ begin{pmatrix}a_1\\2a_2\\3a_3\\0\end{pmatrix}+3\be gin{pmatrix}a_0\\a_1\\a_2\\a_3\end{pmatrix}$