# Problem with change of basis of an polynomial.

Good morning, i have a problem solving this:

Express $a_{0}+a_{1}x+a_{2}x^{2}$ in terms of basis: $1,x-1,x^{2}-1$

I make this:

$c_{1}1+c_{2}(x-1)+c_{3}(x^{2}-1)=c_{1}+c_{2}x-c_{2}+c_{3}x^{2}-c_{3}=0+0x+0x^{2}$

then

$C=\begin{array}{ccc} 1 & 0 & 0\\ -1 & 1 & 0\\ -1 & 0 & 1 \end{array}$

and the matrix change of basis is:

$C^{-1}=\begin{array}{ccc} 1 & 0 & 0\\ -1 & 1 & 0\\ -1 & 0 & 1 \end{array}$

$\Longrightarrow a_{0}(1,-1,-1)+a_{1}x(0,1,0)+a_{2}x^{2}(0,0,1)=a_{0}+a_{1}x+a_{2}x^{2}$

$c_{1}1+c_{2}(x-1)+c_{3}(x^{2}-1)=c_{1}+c_{2}x-c_{2}+c_{3}x^{2}-c_{3}=0+0x+0x^{2}$
$c_{0}1+c_{1}(x-1)+c_{2}(x^{2}-1)=c_{0}+c_{1}x-c_{1}+c_{2}x^{2}-c_{2}=a_0+a_1x+a_2x^{2}$
and calculate $c_0,c_1,c_2$.