# Find the change-of-coordinates matrix from \mathcal{B} to the standard bas

Find the change-of-coordinates matrix from $$\displaystyle{\mathcal{{{B}}}}$$ to the standard basis in $$\displaystyle{\mathbb{{{R}}}}^{{{n}}}$$
$\mathcal{B}=\{\begin{bmatrix}3 \\ -1 \\ 4 \end{bmatrix},\begin{bmatrix}2 \\ 0 \\ -5 \end{bmatrix},\begin{bmatrix}8 \\ -2\\ 7\end{bmatrix}\}$

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Knes1997
Step 1
The change-of-coordinates matrix from $$\displaystyle{\mathcal{{{B}}}}$$ to the standard basis in $$\displaystyle{\mathbb{{{R}}}}^{{{3}}}$$ by the definition is
$P_{B}=\begin{bmatrix} b_{1} & b_{2} & b_{3}\end{bmatrix}=\begin{bmatrix} 3 & 2 & 8 \\ -1 & 0 & -2 \\ 4 & -5 & 7\end{bmatrix}$
$\begin{bmatrix} 3 & 2 & 8 \\ -1 & 0 & -2 \\ 4 & -5 & 7\end{bmatrix}$
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Nancy Johnson
Step 1
The change-of-coordinates matrix from a basis $$\displaystyle{\mathcal{{{B}}}}={\left\lbrace{b}_{{{1}}},\ {b}_{{{2}}},\ \cdots,{b}_{{{n}}}\right\rbrace}$$ to the standard matrix in $$\displaystyle{\mathbb{{{R}}}}^{{{n}}}$$ is given as, $$\displaystyle{P}_{{{\mathcal{{{B}}}}}}={\left[{b}_{{{1}}}\ {b}_{{{2}}}\ \cdots\ {b}_{{{n}}}\right]}$$
Here, n is the number of vectors in a basis.
Step 2
There are three vectors in the given basis,
The given basis is $\mathcal{B}=\{\begin{bmatrix}3 \\ -1 \\ 4 \end{bmatrix},\begin{bmatrix}2 \\ 0 \\ -5 \end{bmatrix},\begin{bmatrix} 8 \\ -2 \\ 7 \end{bmatrix}\}$
Thus, the change-of-coordinates matrix from $$\displaystyle{\mathcal{{{B}}}}$$ to the standard basis in $$\displaystyle{\mathbb{{{R}}}}^{{{3}}}$$ is
$$\displaystyle{P}_{{{\mathcal{{{B}}}}}}={\left[{b}_{{{1}}}\ {b}_{{{2}}}\ {b}_{{{3}}}\right]}$$
$=\begin{bmatrix}3 & 2 & 8 \\ -1 & 0 & -2 \\ 4 & -5 & 7\end{bmatrix}$
Therefore, the change-of-coordinates matrix is $=\begin{bmatrix}3 & 2 & 8 \\ -1 & 0 & -2 \\ 4 & -5 & 7\end{bmatrix}$