Given v=(−5−1−1) v = \begin{pmatrix}-5 \\ -1 \\ -1 \end{pmatrix} v=−5−1−1 and a basis B={(0−43),(434),(−251216)}\mathcal B = \left\{ \begin{pmatrix} 0 \\ -4\\ 3\end{pmatrix}, \begin{pmatrix} 4 \\ 3\\ 4\end{pmatrix}, \begin{pmatrix} -25 \\ 12\\ 16\end{pmatrix} \right\}B=⎩⎨⎧0−43,434,−251216⎭⎬⎫.
v=(−5−1−1) v = \begin{pmatrix}-5 \\ -1 \\ -1 \end{pmatrix} v=−5−1−1
B={(0−43),(434),(−251216)}\mathcal B = \left\{ \begin{pmatrix} 0 \\ -4\\ 3\end{pmatrix}, \begin{pmatrix} 4 \\ 3\\ 4\end{pmatrix}, \begin{pmatrix} -25 \\ 12\\ 16\end{pmatrix} \right\}B=⎩⎨⎧0−43,434,−251216⎭⎬⎫
Calculate the coordinate vector [v]B=(XYZ)[v]_{\mathcal B} = \begin{pmatrix} {\color{red}X} \\ {\color{blue}Y} \\ Z \end{pmatrix}[v]B=XYZ of vvv with respect to the basis B\mathcal B B.
[v]B=(XYZ)[v]_{\mathcal B} = \begin{pmatrix} {\color{red}X} \\ {\color{blue}Y} \\ Z \end{pmatrix}[v]B=XYZ
vvv
B\mathcal B B
X=\color{red} X =X=
Y=\color{blue} Y =Y=
Z=Z =Z=