If the matrix $A$ has a single value decomposition $U\Sigma V^*$, where $^*$ denotes the hermitian conjugate, then the null space of $A$ is the span of the set of vectors $\{ v_{r+1},\dots, v_n\}$, where $r$ denotes the rank of $A$ and $v_j$ denotes the $j$th column of $V$ for $j = 1,\dots, n$.
I'm going through the proof of this in my notes and have gotten to the point that $Ax = 0$ iff $x$ and $v_j$ are orthogonal for $j=1,\dots, r$. My notes from class then say that this is true if and only if $x\in\text{span}\{v_{r+1},\dots,v_{n}\}$