Number of solutions to linear system of equations over GF(2) Linear systems of equations over the reals have either 0, 1 or infinitely many solutions. However, when applied to finite fields (specifically GF(2)), infinitely many is not an option.

Is there a fast general method to calculate the number of distinct solutions to a linear system of equations over GF(2)?

You can assume that Gaussian elimination has already been performed, so an example augmented matrix would be:

$$ \left[
    \begin{array}{ccccccccc|c}1&amp;0&amp;0&amp;0&amp;1&amp;1&amp;1&amp;0&amp;0&amp;1\\
0&amp;1&amp;0&amp;0&amp;1&amp;0&amp;1&amp;0&amp;1&amp;1\\
0&amp;0&amp;1&amp;0&amp;0&amp;1&amp;1&amp;1&amp;0&amp;0\\
0&amp;0&amp;0&amp;1&amp;1&amp;1&amp;1&amp;1&amp;1&amp;1\\
0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0\\
0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0\\
0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0\\
0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0\\
0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0\\
    \end{array}
\right] $$
You can take any subset of the free variables, and then correct any resulting sum using the fixed variables.

Thus, after Gaussian elimination the total number of solutions is  $2^f$ where $f$ is the number of free variables.