# Let B and C be the following ordered bases of R^3

Let B and C be the following ordered bases of ${R}^{3}:$
$B=\left(\left[\begin{array}{c}1\\ 4\\ -\frac{4}{3}\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 8\end{array}\right],\left[\begin{array}{c}1\\ 1\\ -2\end{array}\right]\right)$
$C=\left(\left[\begin{array}{c}1\\ 1\\ -2\end{array}\right],\left[\begin{array}{c}1\\ 4\\ -\frac{4}{3}\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 8\end{array}\right]\right)$ Find the change of coordinate matrix I_{CB}

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$B=\left(\left[\begin{array}{c}1\\ 4\\ -\frac{4}{3}\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 8\end{array}\right],\left[\begin{array}{c}1\\ 1\\ -2\end{array}\right]\right)$
$C=\left(\left[\begin{array}{c}1\\ 1\\ -2\end{array}\right],\left[\begin{array}{c}1\\ 4\\ -\frac{4}{3}\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 8\end{array}\right]\right)$
$\left[\begin{array}{cccccc}1& 1& 0& 1& 0& 1\\ 1& 4& 1& 4& 1& 1\\ -2& -\frac{4}{3}& 8& -\frac{4}{3}& 8& -2\end{array}\right]$
$\left[\begin{array}{cccccc}1& 1& 0& 1& 0& 1\\ 0& 1& \frac{1}{3}& 1& \frac{1}{3}& 0\\ 0& -\frac{2}{3}& 8& -\frac{2}{3}& 8& 0\end{array}\right]\begin{array}{}-{R}_{1}+{R}_{2}\to {R}_{2}\\ 2{R}_{1}+{R}_{3}\to {R}_{3}\\ {R}_{2}×\frac{1}{3}\to \frac{{R}_{2}}{3}\end{array}\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$
$\left[\begin{array}{cccccc}1& 1& 0& 1& 0& 1\\ 0& 1& \frac{1}{3}& 1& \frac{1}{3}& 0\\ 0& 0& 1& 0& 1& 0\end{array}\right]\begin{array}{}-\frac{2}{3}{R}_{2}+{R}_{3}\to {R}_{3}\\ {R}_{3}×\frac{9}{70}\to \frac{{R}_{9{R}_{3}}}{70}\end{array}$
$\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 1\\ 0& 1& 0& 1& 0& 0\\ 0& 0& 1& 0& 1& 0\end{array}\right]\begin{array}{}-\frac{1}{3}{R}_{3}+{R}_{2}\to {R}_{2}\\ -{R}_{2}+{R}_{1}\to {R}_{1}\end{array}$
${I}_{CB}=\left[\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right]$