Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Digital systems, linear algebra, introduction to formal proof / 5 / b

If $x\in \mathrm{Null}(A)$, the $i$th entry of $Ax=0$ says $r_i\cdot x=0$. Any $y\in \mathrm{Row}(A)$ is a linear combination $y={\sum }_i{\alpha }_i{r}_i$, so bilinearity of the dot product givesHenceThis proves the orthogonality for every pair of vectors in the two vector subspaces. In fact the null space equals the orthogonal complement of the row space, since being orthogonal to every row is exactly the system $Ax=0$.