= Solution
The <nullspace> is the vector subspace $\boxed{\ker A=\{x\in\mathbb R^n:Ax=0\}}$. A full real <singular value decomposition> is
$$
\boxed{A=U\Sigma V^T,\quad U\in\mathbb R^{m\times m},\quad \Sigma\in\mathbb R^{m\times n},\quad V\in\mathbb R^{n\times n},}
$$
where $U,V$ are <orthogonal matrices> and $\Sigma_{ij}=0$ for $i\ne j$, with $\Sigma_{jj}=\sigma_j\geq0$ for $1\leq j\leq n$. We may order $\sigma_1\geq\cdots\geq\sigma_n\geq0$. 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>.
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