Combinatorial nature of basis of a vector space?

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Consider an homogenous linear system of equations:

$$Ax=0$$

The vector space of solutions coincides with the null space of A. I'm particularly interested in the case when $A$ is rank deficient, i e the solution can be parameterized.

What is the combinatorial nature of the set of basis ?

Which are the algorithmic procedures available to generate "different" basis of the null space ?