= Solution
If $A^\top x=0$, the <vector> $x$ belongs to the <null space> for which the columns of $L$ are a <basis>. It is therefore a <linear combination> of those columns, so $x=Ly$ for some $y\in\mathbb R^m$. Conversely, each column is annihilated by $A^\top$, hence $A^\top L=0$ and $x=Ly$ implies $A^\top x=0$. Thus
$$
\boxed{A^\top x=0\iff x=Ly\text{ for some }y\in\mathbb R^m.}
$$
The coordinate <vector> $y$ is unique because the columns of $L$ are <linearly independent>.
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