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 / a / Solution

The $m$ columns of $L$ form a basis of $\ker A^{\top }$, so that null space has dimension $m$. The rank-nullity theorem applied to $A^{\top }:{\mathbb{R}}^n\to {\mathbb{R}}^n$ gives $\mathrm{rank}{A}^{\top }=n-m$. A matrix and its transpose have equal matrix rank, hence