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

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

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