How to find the corresponding matrix to the transformation?

Question

Firstly I have tried with writting down a polynomial of the rate 3. Then I derived and antiderived it. And then ? I need to find the matrix which correnspond to the linear transformation T in the standard basis od 2D real spaces.

Answer 1

Apply $\;T\;$ to the given basis elements and write the outcomes as a linear combination of the same basis:

$$\begin{align*}&T1:=\frac4x\int_0^x 1dt-1=4-1=3=&\color{red}3\cdot1+\color{red}0\cdot x+\color{red}0\cdot x^2\\{}\\ &Tx:=\frac4x\int_0^x t\,dt-2x=\frac4x\frac12x^2-2x=0=&\color{red}0\cdot1+\color{red}0\cdot x+\color{red}0\cdot x^2\\{}\\ &Tx^2:=\frac4x\int_0^x t^2dt-3x^2=\frac4{3x}x^3-3x^2=-\frac53x^2=&\color{red}0\cdot1+\color{red}0\cdot x+\color{red}{\left(-\frac53\right)}\cdot x^2\end{align*}$$

and thus the matrix is

$$[T]=\begin{pmatrix}3&0&0\\0&0&0\\0&0&\!\!-\frac53\end{pmatrix}$$