Question

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 \(RR^{n}.\)

Answer 1

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