Consider two different hyperplanes $H$ and $H'$ of $\mathcal{A} =AG(5,q)$ such that $ H \cap H' \ne \emptyset$.
a) Suppose $L$ is a line in $H'$ that is disjoint to $H$. Show that $L$ is parallel to $H\cap H'$ and that $\dim(H\cap H') =3$.
b) Consider the set $\mathcal{H}$ of all affine hyperplanes in $\mathcal{A}$ through $H\cap H'$. How many elements does $\mathcal{H}$ have?
c) Is there a dilation of $\mathcal{A}$ that maps $H$ to $H'$? If so, describe one. If not, explain.
My attempt:
This is so hard. I've never seen these type of questions and have, honestly, no idea where to start.
The only thing I know is that the dimensions of $H$ and $H'$ will be 4. What do I do next? Introduce a basis? What exactly is the intersection of these hyperplanes? How do I count hyperplanes? How in the world can I know whether or not there exists a dilation as described in (c)?
I'm so frustrated and sorry for the hundreds of questions I have. I have my exam in two days and felt really confident prior to opening this past exam paper.
All help is welcome!