Linear Transformation of a Polynomial

Question

I have an operation that takes $ax^2+bx+c$ to $cx^2+bx+a$. I need to find if this corresponds to a linear transformation from $R^3$ to $R^3$, and if so, its matrix.

I know that

$$ ax^2+bx+c = \begin{bmatrix}a&b&c \end{bmatrix} \begin{bmatrix}x^2\\x\\1 \end{bmatrix} $$

If I perform the column operation $C_1 \leftrightarrow C_3$, then I can get the desired result. However, this would mean putting the corresponding elementary matrix to the right of my coefficient matrix like so:

$$ \begin{bmatrix}a&b&c \end{bmatrix} \begin{bmatrix}0&0&1\\0&1&0\\1&0&0 \end{bmatrix} \begin{bmatrix}x^2\\x\\1 \end{bmatrix} = cx^2+bx+a $$

Is this the answer? I have difficulty accepting that the matrix can simply be put in the middle.

Answer 1

I think that what you've made sends $x^2+x+1$ to $cx^2+bx+a$ . Also the transformation should be $\mathbb{R}[x]^{3}\to\mathbb{R}[x]^3$ and not $\mathbb{R}^3\to\mathbb{R}^3$

Answer 2

$$T((ax^2+bx+c)+(a'x^2+b'x+c'))=T((a+a')x^2+(b+b')x+(c+c')) \\=(c+c')x^2+(b+b')x+(a+a') \\=(cx^2+bx+a)+(c'x^2+b'x+a') \\=T(ax^2+bx+c)+T(a'x^2+b'x+c')$$ and linearity is confirmed.