In terms of projective coordinates

$$\lambda \left[ \begin{array}{c}x \\ y \\ 1 \end{array} \right] = ... ...end{array} \right] \left[ \begin{array}{c}X \\ Y \\ Z \\ 1 \end{array} \right]$$

$$\left[ \begin{array}{c}x \\ y \\ 1 \end{array} \right] \in {\cal P}^2$$    and $$\left[ \begin{array}{c}X \\ Y \\ Z \\ 1 \end{array} \right] \in {\cal P}^3$$

are homogeneous coordinates.

The model is linear in projective geometry.