Given a matrix $A\in\mathbb{R}^{m\times n}$ I would like to find a basis for the null space $$ N(A) = \left\{x\in\mathbb{R}^n\,:\, Ax = 0\right\} $$ automatically.
I am looking for a general strategy that I can code into software. I would like, if possible to avoid matrix decompositions, and if possible to specifically avoid computing the SVD. Is there a matrix formula (involving inverses, if needs be) giving a set of basis vectors?