Question # 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}.

Alternate coordinate systems
ANSWERED The change - of - coordinate matrix from $$\mathscr{B} = \left\{\begin{bmatrix}3\\-1\\4\\\end{bmatrix}\begin{bmatrix}2\\0\\ -5 \\\end{bmatrix}\begin{bmatrix}8\\-2\\7\\ \end{bmatrix}\right\}$$
to the standard basis in $$RR^{n}.$$ 2021-02-26
The change - of - coordinates matrix from a basis $$\mathscr{B} = {b_{1}, b_{2},..., b_{n}}$$
to the standadr matrix in $$RR^{n}$$ is given as,
$$P_\mathscr{B}=[b_{1} b_{2},...,b_{n}]$$
Here, n is the number of vectors in a basis.
There are three vectors in the given basis,
The given basis is c$$\mathscr{B} = \left\{\begin{bmatrix}3\\-1\\4\\\end{bmatrix}\begin{bmatrix}2\\0\\ -5 \\\end{bmatrix}\begin{bmatrix}8\\-2\\7\\ \end{bmatrix}\right\}$$.
Thus, the change-of-coordinates matrix from $$\mathscr{B}$$
to the standard basis in $$RR^{3}$$ is,
$$P_{\mathscr{B}} = [b_{1} b_{2} b_{3}]$$
$$\begin{bmatrix}3 & 2 & 8 \\-1 & 0 & -2\\ 4 & -5 & 7 \end{bmatrix}$$
Therefore, the change-of-coordinates matrix is $$\begin{bmatrix}3 & 2 & 8 \\-1 & 0 & -2\\ 4 & -5 & 7 \end{bmatrix}$$