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 / e / Solution

Both $L$ and $R$ have $m$ linearly independent columns. The Gram matrix $R^{\top }R$ is invertible becauseConsequently $R^{\top }$ is a surjective linear map: for any $y\in {\mathbb{R}}^m$, the choice $x=R(R^{\top }R{)}^{-1}y$ satisfies $R^{\top }x=y$. HenceThe latter image of a linear map has dimension $m$, giving $\mathrm{rank}B=m$. If $m=0$, $B=0$ and the conclusion is immediate. This argument explains why the product retains its full possible matrix rank; a general product of two matrices need not do so.