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 / 2 / d / Solution

The reduced row echelon form givesTaking the free variables as $x_2=s$ and $x_4=5t$, the null space isThe two displayed vectors are linearly independent, as their second and fourth coordinates show, and each satisfies $Ax=0$. ThusThis also agrees with the rank-nullity theorem: $4-2=2$.