Let beta = (x^{2} - x, x^{2} + 1,x - 1), beta' = (x^2 - 2x - 3, - 2x^2 + 5x + 5, 2x^2 - x - 3) be ordered bases for P_2(C). Find the change of coordinate matrix Q that changes beta ' -coordinates into beta -coordinates.

Let $\beta =\left({x}^{2}-x,{x}^{2}+1,x-1\right),{\beta }^{\prime }=\left({x}^{2}-2x-3,-2{x}^{2}+5x+5,2{x}^{2}-x-3\right)$ be ordered bases for ${P}_{2}\left(C\right).$ Find the change of coordinate matrix Q that changes ${\beta }^{\prime }$ -coordinates into $\beta$ -coordinates.
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LET A = $\overline{)\begin{array}{cccccc}1& 1& 0& 1& -2& 2\\ -1& 0& 1& -2& 5& -1\\ 0& 1& -1& -3& 5& -3\end{array}}$

It may be observed that the entries in the columns of A are the coefficients of x2, x and the scalar multiples of 1 in the ordered bases β and β’ respectively. To determine the change of coordinates matrix from ${\beta }^{\prime }\to \beta ,$ we will reduce A to its RREF as under: Add 1 times the 1st row to the 2nd row Add -1 times the 2nd row to the 3rd row Multiply the 3rd row by -1/2 Add -1 times the 3rd row to the 2nd row Add -1 times the 2nd row to the 1st row The RREF of A is $\overline{)\begin{array}{cccccc}1& 0& 0& 3& -6& 3\\ 0& 1& 0& -2& 4& -1\\ 0& 0& 1& 1& -1& 2\end{array}}$

Then the change of coordinates matrix from ${\beta }^{\prime }$ to $\beta$ is Q= $\overline{)\begin{array}{ccc}3& -6& 3\\ -2& 4& -1\\ 1& -1& 2\end{array}}$