Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Digital systems, linear algebra, introduction to formal proof / 5 / f / Solution

The reduced row echelon form yieldsTake the free variables in a linear system to be $s=x_2$, $t=x_4$, $u=x_5$. ThenThusEach listed vector lies in the null space, and every solution has the displayed form. Their coordinates in positions 2, 4 and 5 prove linear independence. The dimension also follows from the rank-nullity theorem: $5-2=3$.