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

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

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