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 \(\begin{bmatrix}3 & 2 & 8 \\-1 & 0 & -2\\ 4 & -5 & 7 \end{bmatrix}\)