<s>[INST] Linear Transformation of a Polynomial 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.
 [/INST] $$T((ax^2+bx+c)+(a'x^2+b'x+c'))=T((a+a')x^2+(b+b')x+(c+c'))
\\=(c+c')x^2+(b+b')x+(a+a')
\\=(cx^2+bx+a)+(c'x^2+b'x+a')
\\=T(ax^2+bx+c)+T(a'x^2+b'x+c')$$ and linearity is confirmed.
</s>