Past exam of the numerical linear algebra course of the University of Oxford / 2025 / 2 / a / ii / Solution

Choose the first $r$ left singular vectors:Their orthonormality gives $Q_{\ast }^T{Q}_{\ast }=I_r$. The orthogonal projection $Q_{\ast }{Q}_{\ast }^T$ fixes $u_j$ for $j\le r$ and annihilates $u_j$ for $j>r$. Applying it to the singular value decomposition therefore gives