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 / e

The reduced row echelon form givesChoose the free variables $x_3=s,x_4=t$. ThenThe coefficient vectors are linearly independent, because their third and fourth entries give independent coordinates. ConsequentlySubstitution in the original matrix verifies $Ax=0$ for both basis vectors. The rank-nullity theorem supplies the independent dimension check $2+2=4$.