Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Digital systems, linear algebra, introduction to formal proof (resit) / 6 / a / Solution

For a real $m\times n$ matrix $A$, the row space is the linear span of its rows in ${\mathbb{R}}^n$. The column space is the linear span of its columns in ${\mathbb{R}}^m$, equivalently $\{A\mathbf{x}:\mathbf{x}\in {\mathbb{R}}^n\}$. The null space isThe matrix rank is the dimension of the column space, which equals the dimension of the row space. In particular,The last equality is the rank-nullity theorem.