#mimetex(\text{A} = \left( \begin{array}{c}X\\Y\\Z\\1\end{array} \right))
#mimetex(\text{B} = \left( \begin{array}{c}X^{\prime}\\Y^{\prime}\\Z^{\prime}\\1\end{array} \right))
#mimetex(\text{B} = \text{M}\text{A})
#mimetex(\left( \begin{array}{c}X^{\prime}\\Y^{\prime}\\Z^{\prime}\\1\end{array} \right) = \left(\begin{array}{cccc}M_{11}&M_{12}&M_{13}&M_{14}\\M_{21}&M_{22}&M_{23}&M_{24}\\M_{31}&M_{32}&M_{33}&M_{34}\\0&0&0&1\end{array}\right)\left( \begin{array}{c}X\\Y\\Z\\1\end{array} \right))
#mimetex(\left( \begin{array}{c}X^{\prime}\\Y^{\prime}\\Z^{\prime}\\1\end{array} \right) = \left(\begin{array}{cccc} & \text{R} & & \text{t}\\0&0&0&1\end{array}\right)\left( \begin{array}{c}X\\Y\\Z\\1\end{array} \right))
#mimetex(\left( \begin{array}{c}X^{\prime}\\Y^{\prime}\\Z^{\prime}\end{array} \right) = \left( \begin{array}{cccccccccccc} X&Y&Z&1&0&0&0&0&0&0&0&0\\ 0&0&0&0&X&Y&Z&1&0&0&0&0\\ 0&0&0&0&0&0&0&0&X&Y&Z&1 \end{array} \right)\left(\begin{array}{c} M_{11}\\M_{12}\\M_{13}\\M_{14}\\M_{21}\\M_{22}\\M_{23}\\M_{24}\\M_{31}\\M_{32}\\M_{33}\\M_{34} \end{array}\right))