I'm having a tough time understanding the textbook on how to answer this question? I'm not too sure what to do? Any help will be appreciated.
$$ S=\operatorname{span}\left[ \begin{pmatrix} 1 \\ -3 \end{pmatrix} \right] $$
Could I use any vector that is orthogonal to the span $S$ as the basis?
You approach is correct here. In particular, we can say $$ S^\perp = \text{span}\left\{ \pmatrix{3\\1} \right\} $$