Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2010 / A / 1 / d / Solution

The nullspace is the vector subspace $\boxed{\ker A=\{x\in {\mathbb{R}}^n:Ax=0\}}$. A full real singular value decomposition iswhere $U,V$ are orthogonal matrices and ${\Sigma }_{ij}=0$ for $i\ne j$, with ${\Sigma }_{jj}={\sigma }_j\ge 0$ for $1\le j\le n$. We may order ${\sigma }_1\ge \cdots \ge {\sigma }_n\ge 0$. The columns of $V$ form an orthonormal basis of the input space ${\mathbb{R}}^n$, and the columns of $U$ form one of the output space ${\mathbb{R}}^m$. The positive singular values number exactly the matrix rank.