 # The change - of - coordinate matrix from mathscr{B} = left{begin{bmatrix}3-14end{bmatrix}begin{bmatrix}20 -5 end{bmatrix}begin{bmatrix}8-27 end{bmatrix}right} to the standard basis in RR^{n}. Emily-Jane Bray 2021-02-25 Answered
The change - of - coordinate matrix from $\mathcal{B}=\left\{\left[\begin{array}{c}3\\ -1\\ 4\end{array}\right]\left[\begin{array}{c}2\\ 0\\ -5\end{array}\right]\left[\begin{array}{c}8\\ -2\\ 7\end{array}\right]\right\}$
to the standard basis in $R{R}^{n}.$
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The change - of - coordinates matrix from a basis $\mathcal{B}={b}_{1},{b}_{2},...,{b}_{n}$
to the standadr matrix in $R{R}^{n}$ is given as,
${P}_{\mathcal{B}}=\left[{b}_{1}{b}_{2},...,{b}_{n}\right]$
Here, n is the number of vectors in a basis.
There are three vectors in the given basis,
The given basis is c$\mathcal{B}=\left\{\left[\begin{array}{c}3\\ -1\\ 4\end{array}\right]\left[\begin{array}{c}2\\ 0\\ -5\end{array}\right]\left[\begin{array}{c}8\\ -2\\ 7\end{array}\right]\right\}$.
Thus, the change-of-coordinates matrix from $\mathcal{B}$
to the standard basis in $R{R}^{3}$ is,
${P}_{\mathcal{B}}=\left[{b}_{1}{b}_{2}{b}_{3}\right]$
$\left[\begin{array}{ccc}3& 2& 8\\ -1& 0& -2\\ 4& -5& 7\end{array}\right]$
Therefore, the change-of-coordinates matrix is $\left[\begin{array}{ccc}3& 2& 8\\ -1& 0& -2\\ 4& -5& 7\end{array}\right]$