I'm trying to solve question 9 of page B24 here.
My attempt:
I claim that the possibilites are one of the following
- An empty set.
- A point.
- A line.
For the first case, consider the following example $$x=0,y=0 \text{ and } x=1,z=0 .$$ For the second case, consider $$x=0,y=0 \text{ and } z=0,w=0 $$ For the last case, consider $$x=0,y=0 \text{ and } y=0,z=0 $$
Since the planes are not parallel, they can't coincide, so an intersection of dimension two is impossible. Obviously intersections of higher dimensions are also impossible. Am I correct here? On a different question it is said that an empty intersection is impossible in the non-parallel case, yet I think it is possible.
Thank you!