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Consider the function $T\left(f\right(t\left)\right)=\left[\begin{array}{c}\mathrm{f}\left(5\right)\\ \mathrm{f}\left(7\right)\\ \mathrm{f}\left(11\right)\end{array}\right]from{P}_{2}to{R}^3.$

A function is called a linear transformation on if the function satisfies the following properties.

1. $\mathbf{}\mathit{D}\mathbf{(}\mathit{x}\mathbf{+}\mathit{y}\mathbf{)}\mathbf{=}\mathit{D}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{+}\mathit{D}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{}\mathit{f}\mathit{o}\mathit{r}\mathbf{}\mathit{a}\mathit{l}\mathit{l}\mathbf{}\mathit{x}\mathbf{,}\mathit{y}\mathbf{鈭�}\mathit{R}\mathbf{.}$
   
2. $\mathbf{}\mathit{D}\mathbf{\left(}\mathit{伪}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{伪}\mathit{D}\mathbf{(}\mathit{x}\mathbf{}\mathbf{)}\mathbf{}\mathit{f}\mathit{o}\mathit{r}\mathbf{}\mathit{a}\mathit{l}\mathit{l}\mathbf{}\mathit{c}\mathit{o}\mathit{n}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{t}\mathbf{}\mathit{伪}\mathbf{鈭�}\mathit{R}\mathbf{.}$
   

An invertible linear transformation is called isomorphism or dimension of domain and co-domain is not same then the function is not isomorphism.

Assume $f,g鈭�P_{2}t$then $T\left(f\right(t\left)\right)=\left[\begin{array}{c}\mathrm{f}\left(5\right)\\ \mathrm{f}\left(7\right)\\ \mathrm{f}\left(11\right)\end{array}\right]$and $T\left(g\right(t\left)\right)=\left[\begin{array}{c}g\left(5\right)\\ g\left(7\right)\\ g\left(11\right)\end{array}\right]$.

Substitute the value role="math" localid="1659411768440" $\left[\begin{array}{c}\mathrm{f}\left(5\right)\\ \mathrm{f}\left(7\right)\\ \mathrm{f}\left(11\right)\end{array}\right]forT\left(f\right(t\left)\right)and\left[\begin{array}{c}g\left(5\right)\\ g\left(7\right)\\ g\left(11\right)\end{array}\right]forT\left(g\right(t\left)\right)inT\left(f\right(t\left)\right)+T\left(g\right(t\left)\right)$ as follows.

$T\left(f\right(t\left)\right)+T\left(g\right(t\left)\right)=\left[\begin{array}{c}\mathrm{f}\left(5\right)\\ \mathrm{f}\left(7\right)\\ \mathrm{f}\left(11\right)\end{array}\right]+\left[\begin{array}{c}g\left(5\right)\\ g\left(7\right)\\ g\left(11\right)\end{array}\right]$

Now, simplify $T\left(\left\{f+g\right\}\right(t\left)\right)$as follows.

$$\begin{gathered} T\left(\left\{f+g\right\}\right(t\left)\right)=\left[\begin{array}{cc}\left\{f+g\right\}& \left(5\right)\\ \left\{f+g\right\}& \left(7\right)\\ \left\{f+g\right\}& \left(11\right)\end{array}\right] \\ =\left[\begin{array}{cc}f\left(5\right)& g\left(5\right)\\ f\left(7\right)& g\left(7\right)\\ f\left(11\right)& g\left(11\right)\end{array}\right] \\ =\left[\begin{array}{c}\begin{array}{c}f\left(5\right)\\ f\left(7\right)\\ f\left(11\right)\end{array}\end{array}\right]+\left[\begin{array}{c}\begin{array}{c}g\left(5\right)\\ g\left(7\right)\\ g\left(11\right)\end{array}\end{array}\right] \\ T\left(\left\{f+g\right\}\right(t\left)\right)=T\left(f\right(t\left)\right)+T\left(g\right(t\left)\right) \end{gathered}$$

Assume $f鈭�P_{2}and伪鈭�R$then role="math" localid="1659412647658" $T\left(f\right(t\left)\right)=\left[\begin{array}{c}\begin{array}{c}f\left(5\right)\\ f\left(7\right)\\ f\left(11\right)\end{array}\end{array}\right]$.

Substitute the value $\left[\begin{array}{c}\begin{array}{c}伪f\left(5\right)\\ 伪f\left(7\right)\\ 伪f\left(11\right)\end{array}\end{array}\right]forT\left(伪f\right(t\left)\right)$for as follows.

$$\begin{gathered} T\left(伪f\right(t\left)\right)=\left[\begin{array}{c}\begin{array}{c}伪f\left(5\right)\\ 伪f\left(7\right)\\ 伪f\left(11\right)\end{array}\end{array}\right] \\ =伪\left[\begin{array}{c}\begin{array}{c}f\left(5\right)\\ f\left(7\right)\\ f\left(11\right)\end{array}\end{array}\right] \\ T\left(伪f\right(t)=伪T(f\left(t\right) \end{gathered}$$

As $T(f+g(t\left)\right)=T\left(f\right(t\left)\right)+T\left(g\right(t\left)\right)andT\left(伪f\right(t\left)\right)=伪T\left(f\right(t\left)\right)$, by the definition of linear transformation T is linear.

As the function define from $P_{2}to{R}^{3}and{P}_2$ is spanned by$\left\{1,t,t^2\right\}$means dimension of$P_2$is 3 and dimension of$R^3$is 3.

By the definition of polynomial, every polynomial form $P_2$is described in a unique way means for every$\left[\begin{array}{c}\begin{array}{c}f\left(5\right)\\ f\left(7\right)\\ f\left(11\right)\end{array}\end{array}\right]鈭�R^3$there exist a unique$f\left(t\right)鈭�P_2$such that$T\left(f\right(t\left)\right)=\left[\begin{array}{c}\begin{array}{c}f\left(5\right)\\ f\left(7\right)\\ f\left(11\right)\end{array}\end{array}\right]$.

By the definition of isomorphism, the function is isomorphism.

Hence, the transformationlocalid="1662122176677" $T\left(f\right(t\left)\right)=\left[\begin{array}{c}\begin{array}{c}f\left(5\right)\\ f\left(7\right)\end{array}\end{array}\right]$is linear and isomorphism.