Let $T:P_2(\mathbb{R}) \to P_2(\mathbb{R})$ be a linear transformation such that: $$T(x^2)=3$$ $$T(x)=-3x^2 + 2x -1$$ $$T(1)=4x^2 + 4$$
I already found its associated matrix which is :
$$A=\begin{pmatrix} 0 & -3 & 4 \\ 0 & 2 & 0 \\ 3 & -1 & 4 \\ \end{pmatrix}$$
After finding this matrix, i found the eigenvalues which are: $\lambda_1=-2$,$\lambda_2=2$,$\lambda_3=6$, they are all different so this matrix is diagonalizable and a diagonal matrix could be:
$$D=\begin{pmatrix} -2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 6 \\ \end{pmatrix}$$.
Now comes the problem, the exercise asks me to find a base of $P_2$ with eigenvectors of $T$. I already found the eigenvectors of $T$ associated to each eigenvalue and they form this base:
$$B=\begin{pmatrix} -2/11 & -2 & 2/3 \\ 16/11 & 0 & 0 \\ 1 & 1 & 1 \\ \end{pmatrix}$$ But this base is written as a matrix. How can i write this base but in the form of ''polinomyals''? i dont know if i am making myself clear, sorry in advance.