I don't see how $k < m$ can possibly occur in part (b).
Imagine $X \in R^{n, k}$ describes a matrix for the vectors $x^1$ to $x^K$.
$m = \dim( span(S)) \leq \min(n,k)$
Then if $k < m$ in part (b), then $k < m \leq \min(n,k)$. This inequality chain is impossible.
Imagine n = 100, k = 10.
Then, $10 < m \leq \min(100, 10)$
$10 < m \leq 10$
m cannot be equal to 10, because $10 < 10$ is impossible, and m cannot be less than 10, because $10 < m < 10$ is impossible.
What's going on?