Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Digital systems, linear algebra, introduction to formal proof / 5 / c / ii

The rank-nullity theorem states $\dim V=\dim \ker T+\dim \mathrm{im}T$ for a linear map from a finite-dimensional vector space $V$. Here the domain has dimension four and the image has dimension two. Thus $\dim \ker T=4-2=2$. For example, a kernel basis is $\{(-2,-1,1,0),(-1,-2,0,1)\}$, as follows from the reduced equations.