= Solution
{c}
Retain the first $r$ terms in the <singular value decomposition>:
$$
\boxed{A_r=\sum_{j=1}^r\sigma_ju_jv_j^T
=U\operatorname{diag}(\sigma_1,\ldots,\sigma_r,0,\ldots,0)V^T.}
$$
The <Eckart-Young-Mirsky theorem> says that this <truncated singular value decomposition> is a best <low-rank approximation> in the <spectral norm>. The remaining <singular values> of the difference are $\sigma_{r+1},\ldots,\sigma_n$, together with zeros, so
$$
\boxed{\|A-A_r\|_2=\sigma_{r+1}.}
$$
For $r\geq1$ the displayed <matrix> has <matrix rank> exactly $r$, since every retained <singular value> is positive. For $r=0$, the only <matrix> of <matrix rank> zero is the zero <matrix>, and the same formula gives error $\sigma_1$.
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