Determining the matrix representation of a linear transformation. $P_2$ to $P_2$

Question

Let $T$ be the linear transformation from $P_2$ to $P_2$ defined by : $$T(a+bx+c x^2 )=b-2cx+a x^2 $$

Attempt:

I used the standard basis. $(1,x, x^2 )$

$$T(1)=a$$

$$T(x)=b$$

$$T( x^2 )=-2c$$

Then I get the following matrix:

$$\begin{bmatrix}1&0&0\\0&1&0\\0&0&-2\end{bmatrix}$$

Answer 1

Hint

you have that $$T(a+bx+cx^2 )=b-2cx+ax^2 $$ T is linear so : $$ \implies aT(1)+bT(x)+cT(x^2)=b-2cx+a x^2 $$ To get $T(1)$ you need $a=1, b=c=0$ $$T(1)=x^2$$ To get $T(x)$ you need $a=c=0, b=1$ $$T(x)=1$$ To get $T(x^2)$ you need $a=b=0,c=1$ $$T(x^2)=-2x$$ Therefore $$A=\pmatrix {0&1&0\\0&0&-2\\1&0 &0}$$

Answer 2

$T(1),T(x)$ and $T(x^2)$ are not correct.

$T(1)=x^2$, $T(x)=1$, $T(x^2)=-2x$.

Let $\beta=\{1,x,x^2\}$ then $$[T]_\beta=\begin{pmatrix}0&1&0\\0&0&-2\\1&0&0\end{pmatrix}.$$