Consider two non-intersecting hyperplanes $A,B \subset \mathbb{F}_q^n$ and a line $at+b$, with $a,b \in \mathbb{F}_q^n$, and $t \in \mathbb{F}_q$ intersecting them. Is it always the case that this line intersects A and B exactly once?
This is clearly the case for $n=2$. For $n>2$, this is again clearly the case for special sets of parallel hyperplanes such as fixing any one coordinate. I imagine that since it is true for these the result essentially follows from a change of basis, but I'm having trouble writing it down explicitly
A subspace $U$ of a linear space is itself a linear space. This means that the line spanned by any two points of $U$ must in fact be contained in $U$. Therefore, if a line $\ell$ intersects a hyperplane $U$ of $\mathrm{AG}(n,q)$, then $\ell \subset U$.