Matrix representation of Linear Transformation Polynomial

Question

I've been looking for similar questions in this site but I couldn't find any, so I decided to post my question.

Let a Linear Transformation $T:\mathbb{P_2 \to \mathbb{P_2}}$ be given by the matrix $\begin{bmatrix} -1 & 0 & 1 \\-2 & 1& 0 \\ 1 & -1 & 1\\ \end{bmatrix}$ with respect to the basis $\{ 1+x, x+x^2, 1+x^2 \}$.

Find $T(x-x^2)$. Please help me out of here, and what kind of stuff related to this topic? need reference.

Answer 1

Hint:

1. Write down $x-x^2$ in terms of $1+x$, $x+x^2$ and $1+x^2$. The coefficients needed are the coordinates of $x-x^2$ in that basis. [Hint: one of them is $0$].
2. Multiply the given matrix by the column vector whose entries are the coordinates found. The output is the set of coordinates of $T(x-x^2)$.