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 / d

The pivot columns in the row echelon form are columns 1 and 2. Take the corresponding columns of the original matrix:For a direct check, call these columns $c_1,c_2$. The other columns satisfy $c_3=c_1-c_2$ and $c_4=c_1+c_2$, so they are in the same linear span. Also $c_1,c_2$ are linearly independent: their fourth entries force the coefficient of $c_2$ to vanish, then their first entries force the other coefficient to vanish. Elementary row operations preserve dependence relations among columns, but generally change the column space itself, which is why the original columns must be used.