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 (resit) / 6 / c / Solution

Let $c_1,\dots ,c_4$ be the original columns. The first three are linearly independent: restricting them to rows 2, 3 and 4 gives the triangular matrixwhose determinant is one. Also $c_4=c_1-c_2+c_3$, so a basis for the column space isThese are columns of the original matrix. Taking columns of its reduced row echelon form would generally give the wrong column space.