Past exam of the numerical analysis course of the University of Oxford / 2015 / 3 / d / iii

The orthonormal columns satisfy $\Vert a{\Vert }_2=\Vert b{\Vert }_2=1$ and $a^{T}b=0$. By part (i), $Q_{1}a=e_1$. A product of orthogonal matrices is an orthogonal matrix, since $Q_1^T{Q}_1=I$. ConsequentlyThe last expression is the first coordinate of the transformed second column. Thus the first row isThe one in the first entry uses the nonnegative retained-coordinate convention. A different consistent sign convention could give $(-1,0)$ instead; the second entry must still be zero by preservation of the inner product.