91Ó°ÊÓ

Consider the linear transformation as follows.

$\mathrm{T}:\mathrm{P}→\mathrm{P}$defined as $\mathrm{T}\left(\mathrm{f}\left(\mathrm{t}\right)\right)={\mathrm{f}}^{\text{'}}\left(\mathrm{t}\right)$.

Consider a non zero polynomial as follows.

role="math" localid="1659419688302" $\mathrm{f}\left(\mathrm{t}\right)={\mathrm{a}}_0\mathrm{t}+{\mathrm{a}}_1{\mathrm{t}}^2+{\mathrm{a}}_2{\mathrm{t}}^3+...+{\mathrm{a}}_{\mathrm{n}-1}{\mathrm{t}}^n$, where $0≤\mathrm{i}≤\mathrm{n}-1$there exist at least one ${\mathrm{a}}_{\mathrm{i}}≠0$.

Differentiate the function $\mathrm{f}\left(\mathrm{t}\right)={\mathrm{a}}_0\mathrm{t}+{\mathrm{a}}_1{\mathrm{t}}^2+{\mathrm{a}}_2{\mathrm{t}}^3+...+{\mathrm{a}}_{\mathrm{n}-1}{\mathrm{t}}^n$with respect to $t$as follows.

$$\begin{gathered} \mathrm{f}\left(\mathrm{t}\right)={\mathrm{a}}_0\mathrm{t}+{\mathrm{a}}_1{\mathrm{t}}^2+{\mathrm{a}}_2{\mathrm{t}}^3+...+{\mathrm{a}}_{\mathrm{n}-1}{\mathrm{t}}^n \\ {\mathrm{f}}^{\text{'}}\left(\mathrm{t}\right)={\mathrm{a}}_0+2{\mathrm{a}}_1\mathrm{t}+3{\mathrm{a}}_2{\mathrm{t}}^2+...+{\mathrm{na}}_{\mathrm{n}-1}{\mathrm{t}}^{\mathrm{n}} \end{gathered}$$

Substitute the value${\mathrm{a}}_0+2{\mathrm{a}}_1\mathrm{t}+3{\mathrm{a}}_2{\mathrm{t}}^2+...+n{\mathrm{a}}_{\mathrm{n}-1}{\mathrm{t}}^n$for${\mathrm{f}}^{\text{'}}\left(\mathrm{t}\right)$in$\mathrm{T}\left(\mathrm{f}\left(\mathrm{t}\right)\right)={\mathrm{f}}^{\text{'}}\left(\mathrm{t}\right)$as follows.

localid="1659420089926" $$\begin{gathered} \mathrm{T}\left(\mathrm{f}\left(\mathrm{t}\right)\right)={\mathrm{f}}^{\text{'}}\left(\mathrm{t}\right) \\ \mathrm{T}\left(\mathrm{f}\left(\mathrm{t}\right)\right)={\mathrm{a}}_0+2{\mathrm{a}}_1\mathrm{t}+3{\mathrm{a}}_2{\mathrm{t}}^2+...+{\mathrm{na}}_{\mathrm{n}-1}{\mathrm{t}}^{\mathrm{n}} \end{gathered}$$

Which is also a polynomial.

Thus,$\mathrm{im}\left(\mathrm{T}\right)=\mathrm{P}$

Since, the derivative of a constant is zero.

Therefore,$f\left(t\right)$ is a constant polynomial.

The kernel is as follows.

$\mathrm{Ker}\left(\mathrm{T}\right)=\left\{\mathrm{f}∈\mathrm{P}:\mathrm{f}\left(\mathrm{t}\right)\mathrm{is}\mathrm{a}\mathrm{constant}\mathrm{polunomial}\right\}$

Hence, the image is$im\left(T\right)=\mathrm{P}$ and the kernel is $\mathrm{Ker}\left(\mathrm{T}\right)=\left\{\mathrm{f}∈\mathrm{P}:\mathrm{f}\left(\mathrm{t}\right)\mathrm{is}\mathrm{a}\mathrm{constant}\mathrm{polunomial}\right\}$.