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

prelimaf1 2021-11-21 Answered
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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Expert Answer

Knes1997
Answered 2021-11-22 Author has 566 answers
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
Answered 2021-11-23 Author has 620 answers
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 \[=\begin{bmatrix}3 & 2 & 8 \\ -1 & 0 & -2 \\ 4 & -5 & 7\end{bmatrix}\]
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