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Consider two linear spaces V and W. A function T is said to be linear transformation if the following holds.

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

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

A linear transformation $\mathit{T}\mathbf{:}\mathit{V}\mathbf{鈫�}\mathit{W}$is said to be an isomorphism if and only if$\mathit{k}\mathit{e}\mathit{r}\mathbf{\left(}\mathbf{T}\mathbf{\right)}\mathbf{=}\mathbf{\left\{}\mathbf{0}\mathbf{\right\}}$ and$\mathbf{lm}\mathbf{\left(}\mathbf{T}\mathbf{\right)}\mathbf{=}\mathbf{W}$ or $\mathbf{dim}\mathbf{\left(}\mathbf{V}\mathbf{\right)}\mathbf{=}\mathbf{dim}\mathbf{\left(}\mathbf{W}\mathbf{\right)}$.

The given transformation as follows.

$\mathrm{T}\left(\mathrm{f}\right)=\left[\begin{array}{cc}\mathrm{f}\left(0\right)& \mathrm{f}\left(1\right)\\ \mathrm{f}\left(2\right)& \mathrm{f}\left(3\right)\end{array}\right]$, from ${\mathrm{P}}_2$to ${\mathrm{R}}^{2脳2}$.

By using the definition of linear transformation as follows.

$$\begin{gathered} \mathrm{T}\left(\mathrm{A}+\mathrm{B}\right)=\mathrm{T}\left(\mathrm{A}\right)+\mathrm{T}\left(\mathrm{B}\right) \\ \mathrm{T}\left(\mathrm{kA}\right)=\mathrm{kT}\left(\mathrm{A}\right) \end{gathered}$$

Now, to check the first condition as follows.

Consider the function as follows.

$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}$and$\mathrm{g}\left(\mathrm{x}\right)={\mathrm{mx}}^2+\mathrm{nx}+\mathrm{l}$ be an arbitrary polynomials from ${\mathrm{P}}_2$.

Now, simplify as follows.

$$\begin{gathered} \mathrm{T}\left(\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right)=\mathrm{T}({\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}+\mathrm{mx}+\mathrm{nx}+\mathrm{l}) \\ =\mathrm{T}\left({\mathrm{x}}^2\right(\mathrm{a}+\mathrm{m})+\mathrm{x}(\mathrm{b}+\mathrm{n})+\mathrm{c}+\mathrm{l}) \\ =\left[\begin{array}{cc}\mathrm{c}+\mathrm{l}& (\mathrm{a}+\mathrm{m})+(\mathrm{b}+\mathrm{n})+(\mathrm{c}+\mathrm{l})\\ 4(\mathrm{a}+\mathrm{m})+2(\mathrm{b}+\mathrm{n})+(\mathrm{c}+\mathrm{l})& 9(\mathrm{a}+\mathrm{m})+3(\mathrm{b}+\mathrm{n})+(\mathrm{c}+\mathrm{l})\end{array}\right] \\ =\left[\begin{array}{cc}\mathrm{c}& \mathrm{a}+\mathrm{b}+\mathrm{c}\\ 4\mathrm{a}+2\mathrm{b}+\mathrm{c}& 9\mathrm{a}+3\mathrm{b}+\mathrm{c}\end{array}\right]+\left[\begin{array}{cc}\mathrm{l}& \mathrm{m}+\mathrm{n}+\mathrm{l}\\ 4\mathrm{m}+2\mathrm{n}+\mathrm{l}& 9\mathrm{m}+3\mathrm{n}+\mathrm{l}\end{array}\right] \end{gathered}$$

Simplify further as follows.

$$\begin{gathered} \mathrm{T}\left(\mathrm{f}\right(\mathrm{x})+\mathrm{g}(\mathrm{x}\left)\right)=\left[\begin{array}{cc}\mathrm{c}& \mathrm{a}+\mathrm{b}+\mathrm{c}\\ 4\mathrm{a}+2\mathrm{b}+\mathrm{c}& 9\mathrm{a}+3\mathrm{b}+\mathrm{c}\end{array}\right]+\left[\begin{array}{cc}\mathrm{l}& \mathrm{m}+\mathrm{n}+\mathrm{l}\\ 4\mathrm{m}+2\mathrm{n}+\mathrm{l}& 9\mathrm{m}+3\mathrm{n}+\mathrm{l}\end{array}\right] \\ \mathrm{T}\left(\mathrm{f}\right(\mathrm{x})+\mathrm{g}(\mathrm{x}\left)\right)=\mathrm{T}\left(\mathrm{f}\right(\mathrm{x}\left)\right)+\mathrm{T}\left(\mathrm{g}\right(\mathrm{x}\left)\right) \end{gathered}$$

Thus,T is a linear transformation.

A linear transformation $\mathbf{T}\mathbf{:}\mathbf{V}\mathbf{鈫�}\mathbf{W}$ is isomorphism if and only if$\mathbf{ker}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{\left\{}\mathbf{0}\mathbf{\right\}}$ and $\mathbf{Im}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{W}$

Now, check if$\ker \left(\mathrm{t}\right)=\left\{0\right\}$ as follows.

$\ker \left(\mathrm{T}\right)=\left\{\mathrm{f}\left(\mathrm{x}\right)鈭�{\mathrm{P}}_2|\mathrm{T}\left(\mathrm{f}\left(\mathrm{x}\right)\right)=0\right\}$

Consider the function as follows.

$f\left(x\right)=a{x}^2+bx+c鈭�P_2$be arbitrary polynomial and the set the next equation as follows.

$$\begin{gathered} \mathrm{T}\left(\mathrm{f}\left(\mathrm{x}\right)\right)=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \left[\begin{array}{cc}\mathrm{c}& \mathrm{a}+\mathrm{b}+\mathrm{c}\\ 4\mathrm{a}+2\mathrm{b}+\mathrm{c}& 9\mathrm{a}+3\mathrm{b}+\mathrm{c}\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \end{gathered}$$

Equating the corresponding entries as follows.

$$\begin{gathered} \mathrm{c}=0 \\ \mathrm{a}+\mathrm{b}+\mathrm{c}=0 \\ 4\mathrm{a}+2\mathrm{b}+\mathrm{c}=0 \\ 9\mathrm{a}+3\mathrm{b}+\mathrm{c}=0 \end{gathered}$$

Simplify and solve the equation and the value are as follows.

$\mathrm{a}=\mathrm{b}=\mathrm{c}=0$

A linear transformation$\mathrm{T}:\mathrm{V}鈫�\mathrm{W}$ is an isomorphism only and only if$\ker \left(\mathrm{T}\right)=\left\{0\right\}$ and $\mathrm{lm}\left(\mathrm{T}\right)=\mathrm{W}$.

Now, to check that if $\ker \left(\mathrm{T}\right)=\left\{0\right\}$ as follows.

$\ker \left(\mathrm{T}\right)=\left\{\mathrm{f}\left(\mathrm{x}\right)鈭�{\mathrm{P}}_2|\mathrm{T}\left(\mathrm{f}\left(\mathrm{x}\right)\right)=0\right\}$

Now, to check that if $\mathrm{lm}\left(\mathrm{T}\right)={\mathrm{R}}^{2脳2}$.

Consider the function as follows.

$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}鈭�{\mathrm{P}}_2$be arbitrary polynomial and the set the next equation as follows.

$\begin{array}{l}\mathrm{T}\left(\mathrm{f}\right(\mathrm{x}\left)\right)=\left[\begin{array}{cc}\mathrm{c}& \mathrm{a}+\mathrm{b}+\mathrm{c}\\ 4\mathrm{a}+2\mathrm{b}+\mathrm{c}& 9\mathrm{a}+3\mathrm{b}+\mathrm{c}\end{array}\right]\\ \mathrm{T}\left(\mathrm{f}\right(\mathrm{x}\left)\right)=\mathrm{B}\end{array}$

For each matrix B do not exist polynomial f(x) so that T(f ( x )) = B holds.

This implied as follows.

role="math" localid="1659857170557" ${\mathrm{lmmR}}^{2脳2}$

Since,$\ker \left(\mathrm{T}\right)=\left\{0\right\}$ and$\mathrm{Im}\left(\mathrm{T}\right)鈮�{\mathrm{P}}_2$

Thus,T is not an isomorphism.

Hence, T is a linear transformation also$\ker \left(\mathrm{T}\right)=\left\{0\right\}$ and $\mathrm{Im}\left(\mathrm{T}\right)鈮�{\mathrm{P}}_2$, then not an isomorphism.