Consider the following maps of polynomials \(S:\mathcal{P}\rightarrow\mathcal{P}\) and \(T:\mathcal{P}\rightarrow\mathcal{P}\) defined by

\[ S(g(x))= -5 \, x^{3} g\left(x\right) - 4 \, g'\left(x\right) \hspace{1em} \text{and} \hspace{1em} T(g(x))= 5 \, g\left(x\right)^{2} - 3 \, g\left(1\right) \]

Explain why one these maps is a linear transformation and why the other map is not.