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 / b / Solution

Let the original rows be $r_1,\dots ,r_5$. Begin with rows $r_2$, $r_3-r_2$ and $r_4$:Subtract the third from the second, then subtract that new second row from the first. The resulting rows areThey are linearly independent because their first three coordinates form the identity matrix. All original rows lie in their span: $r_1=2r_2+r_4$ and $r_5=r_2-(r_3-r_2)+r_4$. Thus they form a basis of the row space, and $\mathrm{rank}A=3$. Equivalently, these are the nonzero rows of the reduced row echelon form, with two further zero rows.