Past exam of the options II course of the University of Oxford / 2008 / P / P4 / b / Solution

Only the first and third coordinates occur in the first two columns. Normalize the first column to get $q_1=(4,0,3,0{)}^T/5$. Removing its component from the second column leaves $(21,0,-28,0{)}^T/5$, of length seven, so $q_2=(3,0,-4,0{)}^T/5$. The remaining orthonormal columns can be $q_3=e_2$ and $q_4=e_4$. Taking their dot products with the columns of $A$ givesThe columns of $Q$ are orthonormal, and direct multiplication gives the source matrix. Exploiting its two-coordinate first block avoids unnecessary four-dimensional elimination.