Calculating the fundamental matrix

Equation (3) can be written as

$${\bf U}^{\top}{\bf f} = 0,$$

where

${\bf U} = [uu', vu', u', uv', vv', v', u, v, 1]$ and
${\bf f} = [F_{11}, F_{12}, F_{13}, F_{21}, F_{22}, F_{23}, F_{31}, F_{32}, F_{33}]$.

This equation is linear and homogeneous in the 9 unknown coefficients of the matrix F. Thus, if we are given 8 matches we will, in general, be able to determine a unique solution to F, defined up to a scale factor.