<s>[INST] Exercise about basis I am trying to solve the following exercise:
Let $A \in L(P_3)$ be defined by a matrix:
where $A^b_e$ =  $\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}$ (The transition matrix from basis $b$ to basis $e$)
where $e= \{1,x,x^2\}$ and $b=\{1-x, 2x, x+x^2\}$. Find all the $p\in P_3$ for which $(Ap)(x)=2+x+x^2$.
How I solve it:
I have started with calculating the inverse matrix of $b$:
I have found
$\begin{pmatrix} 1 & 0 & 0 \\ 1/2 & 1/2 & -1/2 \\ 0 & 0 & 1 \end{pmatrix}$
secondly, I multiply the two matrices.
$\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 0 & 0 \\ 1/2 & 1/2 & -1/2 \\ 0 & 0 & 1 \end{pmatrix}= \begin{pmatrix} 1 & 0 & 1 \\ 1/2 & 1/2 & -1/2 \\ 3/2 & 1/2 & -1/2 \end{pmatrix}$
and my last step to find all the polynomials matching.
$$ \left[
      \begin{array}{ccc|c}
        1&0&1&2\\
        1/2&1/2&-1/2&1\\
        1/2&1/2&-1/2&1
      \end{array}
    \right]$$
But I don't know if I am right and how to continue. Can someone help me please?
 [/INST] I would start by solving the equivalent system in $\mathscr{F}^3$ (so working with the matrix and coordinate vectors only) - so we have to solve: \begin{equation} \left[
      \begin{array}{ccc|c}
        1&0&1&2\\
        0&1&1&1\\
        1&1&1&1
      \end{array}
    \right]. \end{equation} So then we get the (unique) coordinate vector (we would expect only one solution since any change of basis matrix is invertible): \begin{equation} \begin{bmatrix} 0 \\ -1 \\2 \end{bmatrix}_b, \end{equation}as solution so that we have $p=-(2x)+2(x+x^2)=-2x+2x+2x^2=2x^2$.  
</s>