91Ó°ÊÓ

Consider two linear spaces V and W. A transformation T is said to be a linear transformation if it satisfies the properties,

$$\begin{gathered} \mathit{T}\mathbf{(}\mathbf{f}\mathbf{+}\mathbf{g}\mathbf{)}\mathbf{=}\mathit{T}\mathbf{\left(}\mathbf{f}\mathbf{\right)}\mathbf{+}\mathit{T}\mathbf{(}\mathbf{g}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{T}\mathbf{\left(}\mathbf{kv}\mathbf{\right)}\mathbf{=}\mathit{k}\mathit{T}\mathbf{\left(}\mathbf{v}\mathbf{\right)} \end{gathered}$$

For all elements f,g of v and k is scalar.

An invertible linear transformation is called an isomorphism.

Consider the transformation $T\left(\mathrm{f}\left(\mathrm{t}\right)\right)=F\text{'}\text{'}\left(\mathrm{t}\right)f\left(\mathrm{t}\right)$, from$P_2$ to $P_2$.

Check whether the transformation T satisfies the below two conditions or not.

$$\begin{gathered} 1.T(f+g)=T\left(f\right)+T\left(g\right) \\ 2.T\left(kf\right)=kT\left(f\right) \end{gathered}$$

Verify the first condition.

Let$f\left(t\right)=a_1{t}^2+b_{1}t+c_1$and$g\left(t\right)=a_2{t}^2+b_{2}t+c_2$be two polynomial functions from $P_2$. Then,$T\left(\mathrm{f}\left(\mathrm{t}\right)\right)=F\text{'}\text{'}\left(\mathrm{t}\right)f\left(\mathrm{t}\right)$,$T\left(g\left(\mathrm{t}\right)\right)=g\text{'}\text{'}\left(\mathrm{t}\right)g\left(\mathrm{t}\right)$

Find $T\left(f\left(t\right)+g\left(t\right)\right)$.

$$\begin{gathered} T\left(\left(f+g\right)\left(t\right)\right)=\left(f+g\right)\text{'}\text{'}\left(f+g\right) \\ =\left(f\text{'}\text{'}+g\text{'}\text{'}\right)\left(f+g\right) \\ =f\text{'}\text{'}f+g\text{'}\text{'}f+f\text{'}\text{'}g+g\text{'}\text{'}g \\ =f\text{'}\text{'}f+g\text{'}\text{'}g+g\text{'}\text{'}f+f\text{'}\text{'}g \\ =T\left(f\left(t\right)\right)+T\left(g\left(t\right)\right)+g\text{'}\text{'}f+f\text{'}\text{'}g \\ T\left(\left(f+g\right)\left(t\right)\right)≠T\left(f\left(t\right)\right)+T\left(g\left(t\right)\right) \end{gathered}$$

It is clear that, the first condition$T\left(f+g\right)=T\left(f\right)+T\left(g\right)$ is not satisfied.

Thus, T is not a linear transformation.