Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Digital systems, linear algebra and introduction to formal proof / 4 / b

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 $A^{\top }x=0$. ThusThe coordinate vector $y$ is unique because the columns of $L$ are linearly independent.