coordinate transformation

$X_{0}$为$I$在$O_{0}$系的坐标${\left(
\begin{array}{c}
x_0 \\
y_0 \\
z_0 \\
\end{array}
\right)}$,$X_{1}$为$I$在$O_{1}$系的坐标${\left(
\begin{array}{c}
x_1 \\
y_1 \\
z_1 \\
\end{array}
\right)}$

$X_{1}=AX_{0}+B$

\(\overset{\rightharpoonup }{i}_0\)=\(a_{11}\)\(\overset{\rightharpoonup }{i}_1\)+\(a_{21}\)\(\overset{\rightharpoonup }{j}_1\)+\(a_{31}\)\(\overset{\rightharpoonup
}{k}_1\)

\(\overset{\rightharpoonup }{j}_0\)=\(a_{11}\)\(\overset{\rightharpoonup }{i}_1\)+\(a_{21}\)\(\overset{\rightharpoonup }{j}_1\)+\(a_{31}\)\(\overset{\rightharpoonup
}{k}_1\)

\(\overset{\rightharpoonup }{k}_0\)=\(a_{11}\)\(\overset{\rightharpoonup }{i}_1\)+\(a_{21}\)\(\overset{\rightharpoonup }{j}_1\)+\(a_{31}\)\(\overset{\rightharpoonup
}{k}_1\)

\(\overset{\rightharpoonup }{O_0O_1}\)  在系$O_{1}$中的坐标为

${-B=\left(
\begin{array}{c}
b_1 \\
b_2 \\
b_3 \\
\end{array}
\right)}$

\(\overset{\rightharpoonup }{O_0I}\)  在系$O_{0}$中的坐标为

${X_0=\left(
\begin{array}{c}
x_0 \\
y_0 \\
z_0 \\
\end{array}
\right)}$

\(\overset{\rightharpoonup }{O_0I}\)  在系$O_{1}$中的坐标为

${X_1-B=\left(
\begin{array}{c}
x_1 \\
y_1 \\
z_1 \\
\end{array}
\right)-\left(
\begin{array}{c}
b_1 \\
b_2 \\
b_3 \\
\end{array}
\right)}$

$A=\begin{pmatrix}
a_{11} & a_{12} &a_{13} \\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}
\end{pmatrix}$

${\left(\overset{\rightharpoonup }{i_1},\overset{\rightharpoonup }{j_1},\overset{\rightharpoonup }{k_1}\right)\left(\left(
\begin{array}{c}
x_1 \\
y_1 \\
z_1 \\
\end{array}
\right)-\left(
\begin{array}{c}
b_1 \\
b_2 \\
b_3 \\
\end{array}
\right)\right)=}\\
\pmb{\left(\overset{\rightharpoonup }{i_1},\overset{\rightharpoonup }{j_1},\overset{\rightharpoonup }{k_1}\right)\left(
\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33} \\
\end{array}
\right)\left(
\begin{array}{c}
x_0 \\
y_9 \\
z_0 \\
\end{array}
\right)}$

即

${\left(\left(
\begin{array}{c}
x_1 \\
y_1 \\
z_1 \\
\end{array}
\right)-\left(
\begin{array}{c}
b_1 \\
b_2 \\
b_3 \\
\end{array}
\right)\right)=\left(
\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33} \\
\end{array}
\right)\left(
\begin{array}{c}
x_0 \\
y_9 \\
z_0 \\
\end{array}
\right)}$

即

${X_0-B=A X_1}$

其中

${-B=\left(
\begin{array}{c}
b_1 \\
b_2 \\
b_3 \\
\end{array}
\right)}$

${X_0=\left(
\begin{array}{c}
x_0 \\
y_0 \\
z_0 \\
\end{array}
\right)}$

${X_1=\left(
\begin{array}{c}
x_1 \\
y_1 \\
z_1 \\
\end{array}
\right)}$

$A=\begin{pmatrix}
a_{11} & a_{12} &a_{13} \\ 
a_{21} & a_{22} & a_{23}\\ 
a_{31} & a_{32} & a_{33}
\end{pmatrix}$