Why is DTIME(n) not equal to NP, and consequently, DSPACE(n) not equal to NP?
$DTIME(n^2) \subseteq \mathcal{P} \subseteq \mathcal{NP}$ and by the Time Hierarchy Theorem  $DTIME(n^2) \nsubseteq DTIME(n)$
Therefore $\mathcal{NP} \neq DTIME(n)$