A1 - Linear maps


Example 1 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 2 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 3 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 4 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 5 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 6 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 7 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 8 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 9 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 10 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 11 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 12 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 13 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 14 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 15 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 16 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 17 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 18 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 19 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 20 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 21 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 22 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 23 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 24 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 25 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 26 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 27 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 28 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 29 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 30 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 31 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 32 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 33 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 34 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 35 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 36 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 37 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 38 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 39 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 40 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 41 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 42 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 43 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 44 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 45 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 46 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.


Example 47 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 48 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 49 πŸ”—

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.

Answer:

\(S\) is linear and \(T\) is not linear.


Example 50 πŸ”—

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.

Answer:

\(S\) is not linear and \(T\) is linear.