Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Digital systems, linear algebra, introduction to formal proof (resit) / 6 / a

For an $m\times n$ matrix, its row space is the span of its rows, a subspace of ${\mathbb{R}}^n$; its column space is the span of its columns, a subspace of ${\mathbb{R}}^m$. Its null space is $\ker A=\{x\in {\mathbb{R}}^n:Ax=0\}$. The rank of a matrix is the dimension of its column space, equal to the dimension of its row space. The rank-nullity theorem gives $\dim \ker A=n-\mathrm{rank}A$.