Converting from standard coordinates to general coordinates: Consider these two ordered bases of \(\RNrSpc{3}\):
\[ \begin{array}{llll} \OrdVSpcBss{S}\DefEq (\StdBssVec{1},\StdBssVec{2},\StdBssVec{3})\quad & \StdBssVec{1}\DefEq (1,0,0) & \StdBssVec{2} \DefEq (0,1,0) & \StdBssVec{3} \DefEq (0,0,1) \\ \OrdVSpcBss{B}\DefEq (\Vect{b}_1,\Vect{b}_2,\Vect{b}_3)\quad & \Vect{b}_1 \DefEq (2,-3,7) & \Vect{b}_2 \DefEq (-1,9,2) & \Vect{b}_3 \DefEq (4,2,-6) \end{array} \]Find the matrix \(\CoordTrafoMtrx{C}{B}{S}\) which converts from \(\OrdVSpcBss{S}\)-coordinates to \(\OrdVSpcBss{B}\)-coordinates. Then find \(\CoordVect{x}{B}\) if \(\CoordVect{x}{S} = (4,-1,2)\).
From the information given, we know immediately that the column vectors of \(\CoordTrafoMtrx{C}{S}{B}\) consist of the coordinate vectors of \(\Vect{b}_1\), \(\Vect{b}_2\), and \(\Vect{b}_3\) with respect to \(\EuScript{S}\). As \(\EuScript{S}\) is the standard basis of \(\RNrSpc{3}\), we have
| \((\Vect{b}_1)_{\OrdVSpcBss{S}}\) | \(=\) | \(\Vect{b}_1 = (2,-3,7)\) |
| \((\Vect{b}_2)_{\OrdVSpcBss{S}}\) | \(=\) | \(\Vect{b}_2 = (-1,9,2)\) |
| \((\Vect{b}_3)_{\OrdVSpcBss{S}}\) | \(=\) | \(\Vect{b}_3 = (4,2,-6)\) |
Using these coordinate vectors as the column vectors of the coordinate conversion matrix \(\CoordTrafoMtrx{C}{S}{B}\), we obtain
\[ \CoordTrafoMtrx{C}{S}{B}\ =\ \left[\begin{array}{rrr} 2 & -1 & 4 \\ -3 & 9 & 2 \\ 7 & 2 & -6 \end{array}\right] \]Consequently,
\[ \CoordTrafoMtrx{C}{B}{S} = \left( \CoordTrafoMtrx{C}{S}{B}\right)^{-1}\ =\ \left[\begin{array}{rrr} 2 & -1 & 4 \\ -3 & 9 & 2 \\ 7 & 2 & -6 \end{array}\right]^{-1}\ =\ \frac{1}{388} \left[\begin{array}{rrr} 58 & -2 & 38 \\ 4 & 40 & 16 \\ 69 & 11 & -15 \end{array}\right] \]To find the \(\OrdVSpcBss{B}\)-coordinates of \(\CoordVect{x}{S}\), we compute
\[ \CoordVect{x}{B}\ =\ \frac{1}{388} \left[\begin{array}{rrr} 58 & -2 & 38 \\ 4 & 40 & 16 \\ 69 & 11 & -15 \end{array}\right] \left[\begin{array}{r} 4 \\ -1 \\ 2 \end{array}\right]\ =\ \frac{1}{388} \left[\begin{array}{r} 310 \\ 8 \\ 235 \end{array}\right] \]