The basis of the null space will be the vectors spanned by the basic solutions of the solution to AX = 0. In terms of linear transformations, the null space of a transformation matrix A is the set of all vectors that are transformed to the zero vector.
For example, suppose we have the matrix and its reduced form
The fourth variable is the non-leading, and can be replaced by the parameter s, yielding a basic solution.
Since s is a free variable, this is the same as