= Solution
{c}
Write $r=(r_1,\ldots,r_n)^T$. The <linear map> represented by $A$ has the form
$$
Ax=(r^Tx)e_i.
$$
The <image of a linear map> here is contained in the one-dimensional <linear span> of $e_i$, so its <matrix rank> is at most one. The <rank-nullity theorem> yields
$$
\boxed{\dim\ker A=n-\operatorname{rank}A\ge n-1.}
$$
Thus the solution space of $Ax=0$, namely the zero <eigenspace> when zero is an <eigenvalue>, contains at least $n-1$ <linearly independent vectors>. If $r=0$ its <dimension> is $n$; otherwise it is $n-1$. When $n=1$ the asserted lower bound is zero, even if zero is not an <eigenvalue>.
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