Computing Linear operator on the basis B.

Compute [T]_B and determine whether B is a basis consisting of eigenvectors of T.

a) V= P_2(R), T(a+bx+cx^2)= (-4a+2b-2c)-(7a+3b+7c)x +(7a+b+5c)x^2,

and a basis B= {x-x^2, -1+X^2, -1-x+x^2}

b)V=M_2x2(R), $\displaystyle \begin{pmatrix}\ a & b \\ c & d\end{pmatrix}\$

= $\displaystyle \begin{pmatrix}\-7a-4b+4c-4d & b\\-8a-4b+5c-4d & d\end{pmatrix}\$

And the basis B = {$\displaystyle \begin{pmatrix}\ 1 & 0 \\ 1 & 0\end{pmatrix}$ $\displaystyle \begin{pmatrix}\ -1 & 2 \\ 0 & 0\end{pmatrix}$ $\displaystyle \begin{pmatrix}\ 1 & 0 \\ 2 & 0\end{pmatrix}$ $\displaystyle \begin{pmatrix}\ -1 & 0 \\ 0 & 2\end{pmatrix}$}