## A1 - Linear maps

#### Example 1 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(h(x))= -3 \, x^{2} + 4 \, h\left(x^{3}\right) & \text{and} & T(h)= x^{2} h\left(x\right) + 2 \, h'\left(1\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 2 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(h(x))= -2 \, h\left(2\right) + 5 \, h\left(x^{2}\right) & \text{and} & T(h)= -h\left(x\right) h'\left(x\right) + 2 \, h'\left(1\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 3 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(h(x))= 5 \, h\left(x^{2}\right) - 3 \, h'\left(x\right) & \text{and} & T(h)= 3 \, x^{2} h\left(x\right) + 5 . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 4 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(f(x))= -5 \, f\left(x\right)^{2} - 5 \, f\left(5\right) & \text{and} & T(f)= 5 \, f'\left(1\right) + 5 \, f'\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 5 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(h(x))= -x^{3} h\left(x\right) + 3 \, h\left(x\right) h'\left(x\right) & \text{and} & T(h)= 5 \, h\left(x\right) - 2 \, h'\left(-1\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 6 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(f(x))= 3 \, f\left(1\right) + 3 \, f'\left(x\right) & \text{and} & T(f)= 2 \, x + 3 \, f'\left(5\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 7 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(f(x))= 5 \, f\left(x\right) + f'\left(x\right) & \text{and} & T(f)= 3 \, x^{2} + 4 \, f'\left(1\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 8 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(h(x))= 3 \, h\left(x\right) h'\left(x\right) - 4 \, h\left(-4\right) & \text{and} & T(h)= 5 \, x h\left(x\right) - 3 \, h'\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 9 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= 2 \, g\left(x\right) g'\left(x\right) + g'\left(x\right) & \text{and} & T(g)= -3 \, x^{2} g\left(x\right) - g\left(-3\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 10 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(f(x))= 3 \, x^{2} f\left(x\right) - 4 \, f\left(1\right) & \text{and} & T(f)= -2 \, f\left(x\right) f'\left(x\right) - 3 \, f'\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 11 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(h(x))= -2 \, h\left(2\right) + h'\left(x\right) & \text{and} & T(h)= 5 \, x h\left(x\right) - 5 . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 12 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= -2 \, g\left(x\right)^{2} - 4 \, g\left(-4\right) & \text{and} & T(g)= 2 \, g\left(x^{3}\right) + 3 \, g'\left(5\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 13 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= -5 \, g\left(x\right) g'\left(x\right) + 3 \, g\left(x\right) & \text{and} & T(g)= g\left(3\right) + 5 \, g'\left(-1\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 14 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(f(x))= -x f\left(x\right) - 4 \, f\left(3\right) & \text{and} & T(f)= 5 \, x^{3} + 2 \, f'\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 15 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= 4 \, g\left(x\right) g'\left(x\right) + 4 \, g\left(5\right) & \text{and} & T(g)= x^{3} g\left(x\right) - g'\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 16 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(h(x))= -2 \, x h\left(x\right) - h\left(x\right) & \text{and} & T(h)= h'\left(2\right) - 1 . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 17 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= -4 \, g\left(x\right) g'\left(x\right) - 3 \, g'\left(x\right) & \text{and} & T(g)= g\left(5\right) + 3 \, g'\left(1\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 18 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(f(x))= -2 \, x f\left(x\right) + 5 \, f'\left(-4\right) & \text{and} & T(f)= -4 \, x^{2} + 3 \, f'\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 19 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(f(x))= -3 \, f\left(x\right) + f'\left(x\right) & \text{and} & T(f)= -5 \, f\left(x\right)^{2} + 3 \, f\left(3\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 20 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(h(x))= -2 \, x^{3} h\left(x\right) - h\left(x^{2}\right) & \text{and} & T(h)= 5 \, h\left(x\right)^{2} + 4 \, h'\left(-2\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 21 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= -4 \, x^{2} + 4 \, g\left(x\right) & \text{and} & T(g)= -3 \, g\left(-1\right) - g'\left(-2\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 22 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(f(x))= 2 \, x^{3} f\left(x\right) - 3 \, f\left(x^{3}\right) & \text{and} & T(f)= 5 \, f\left(x\right) f'\left(x\right) + 2 \, f\left(2\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 23 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(f(x))= -f'\left(-2\right) - 2 & \text{and} & T(f)= f\left(x\right) - 5 \, f'\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 24 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= -4 \, x^{2} g\left(x\right) + 5 \, g\left(1\right) & \text{and} & T(g)= g\left(x\right) g'\left(x\right) - 3 \, g'\left(-3\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 25 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(h(x))= 4 \, h\left(3\right) + 4 \, h'\left(3\right) & \text{and} & T(h)= -2 \, h\left(x\right)^{2} - 5 \, h\left(x^{2}\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 26 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= 2 \, g'\left(-1\right) + 4 \, g'\left(x\right) & \text{and} & T(g)= 3 \, x^{2} g\left(x\right) - g\left(x\right) g'\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 27 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(f(x))= -f\left(-3\right) + f'\left(x\right) & \text{and} & T(f)= -2 \, f\left(x\right) f'\left(x\right) + 4 \, f\left(x^{3}\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 28 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= x^{2} g\left(x\right) - 2 \, g'\left(-2\right) & \text{and} & T(g)= 2 \, x^{2} - 5 \, g'\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 29 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(h(x))= -4 \, x^{2} - h\left(5\right) & \text{and} & T(h)= 5 \, h\left(x\right) - 5 \, h'\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 30 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(f(x))= 4 \, x^{3} f\left(x\right) - 3 \, f\left(x\right) f'\left(x\right) & \text{and} & T(f)= -4 \, f'\left(-1\right) - 3 \, f'\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 31 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= -4 \, x^{3} - 3 \, g\left(-1\right) & \text{and} & T(g)= 2 \, g\left(x\right) - 3 \, g'\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 32 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(h(x))= -x h\left(x\right) + 2 \, h'\left(x\right) & \text{and} & T(h)= -3 \, h\left(x\right) h'\left(x\right) + 3 \, h\left(-3\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 33 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(h(x))= -x h\left(x\right) + h'\left(x\right) & \text{and} & T(h)= -4 \, h\left(x\right) h'\left(x\right) - 3 \, h\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 34 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(f(x))= f\left(-3\right) + f'\left(3\right) & \text{and} & T(f)= 3 \, f\left(x\right)^{3} - 4 \, f\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 35 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(h(x))= -2 \, h\left(x\right)^{2} + 5 \, h\left(-5\right) & \text{and} & T(h)= -4 \, h\left(x^{3}\right) - 3 \, h'\left(-4\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 36 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= 3 \, g\left(-4\right) - 2 \, g'\left(x\right) & \text{and} & T(g)= 2 \, g\left(x\right) g'\left(x\right) - g'\left(4\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 37 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(h(x))= -h\left(x\right)^{3} + h\left(1\right) & \text{and} & T(h)= 5 \, h'\left(-5\right) + 3 \, h'\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 38 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= -x^{2} - 5 \, g'\left(-1\right) & \text{and} & T(g)= 4 \, x g\left(x\right) - 4 \, g'\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 39 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(f(x))= x^{2} f\left(x\right) - 4 \, f'\left(4\right) & \text{and} & T(f)= 3 \, f\left(x\right) f'\left(x\right) + 4 \, f'\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 40 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= 2 \, g\left(x\right) g'\left(x\right) + 4 \, g\left(x\right) & \text{and} & T(g)= 3 \, g\left(1\right) + g'\left(2\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 41 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(f(x))= 3 \, x^{3} f\left(x\right) + 2 \, f\left(1\right) & \text{and} & T(f)= -3 \, f\left(x\right) f'\left(x\right) - 3 \, f'\left(5\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 42 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(f(x))= f\left(x\right) f'\left(x\right) - f\left(x\right) & \text{and} & T(f)= -4 \, f\left(-3\right) + 3 \, f'\left(x\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 43 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= 4 \, x^{2} g\left(x\right) + g'\left(4\right) & \text{and} & T(g)= x + 5 \, g\left(2\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 44 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= -2 \, g\left(2\right) + 2 \, g'\left(x\right) & \text{and} & T(g)= -2 \, g\left(x\right)^{2} - 2 \, g'\left(4\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 45 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= -g\left(x\right)^{2} + 2 \, g'\left(x\right) & \text{and} & T(g)= -4 \, g\left(3\right) - 2 \, g'\left(-5\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 46 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(f(x))= 3 \, x^{3} f\left(x\right) + 2 \, x & \text{and} & T(f)= 4 \, f\left(x^{2}\right) - 5 \, f'\left(-1\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is not linear and $$T$$ is linear.

#### Example 47 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= -2 \, g\left(x^{3}\right) - 2 \, g\left(x\right) & \text{and} & T(g)= 3 \, g\left(x\right)^{2} - g'\left(4\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 48 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(g(x))= 2 \, g'\left(-5\right) + 2 \, g'\left(x\right) & \text{and} & T(g)= 2 \, g\left(x\right) g'\left(x\right) - 4 \, g\left(-3\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.

#### Example 49 🔗

Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(f(x))= -5 \, x^{2} f\left(x\right) - 2 \, f'\left(5\right) & \text{and} & T(f)= -4 \, f\left(x\right)^{2} - 2 \, f\left(5\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.

$$S$$ is linear and $$T$$ is not linear.
Consider the following maps of polynomials $$S:\mathcal{P}\rightarrow\mathcal{P}$$ and $$T:\mathcal{P}\rightarrow\mathcal{P}$$ defined by \begin{align*} S(h(x))= -3 \, x^{3} + 5 \, h\left(x^{2}\right) & \text{and} & T(h)= 4 \, x h\left(x\right) + 4 \, h'\left(-5\right) . \\ \end{align*} Explain why one these maps is a linear transformation and why the other map is not.
$$S$$ is not linear and $$T$$ is linear.