I'm looking for an algorithm to solve for the classic:

$$A\mathbf{x} = \mathbf{b}$$

I cannot compute $A$ directly, but rather can compute matrix-vector products $A\mathbf{v}$ for any $\mathbf{v}$.

At first, this appears to be a straightforward application of matrix-free solvers. Since I don't have any guarantees on symmetry or positive-definitness, this seems like a good application of the BICGSTAB algorithm.

The only problem is that I am interested in the special case where $\mathbf{b} = \mathbf{0}$, i.e. solving for a vector in the nullspace of $A$. Obviously, the trivial vector $\mathbf{x}$ = $\mathbf{0}$ is a solution, but not an interesting or useful one. Unfortunately, it's also one that matrix-free methods are prone to find. (It's also the min-norm solution, which I think is problematic as some solutions search for the min-norm solution.)

Is there any trick I can use to extract a non-trivial solution $\mathbf{x}$ with a matrix-free solver for general $A$?

(Cross-post from math.stackexchange.com; I think this problem is a little more research-y than I originally thought)