Let $A$ be a affine space based on a vector space $V$ over a division ring $K$ such that $|K|\neq 2$. Let $U , W$ be two affine subspaces such that $U\subset W$ and $dim(U)=i$, $dim(W)=i+1$, then show that there exists two distinct points $P, Q \in W-U$ such that the line joining $P$ and $Q$ intersects the space $U$.
My attempt: If not, then for every two distinct points in $W-U$, the line joining those two points lie in $W-U$, which makes $W-U$, a affine subspace. But now, I am struck at this point! I think $W-U$ cannot be a affine subspace, but couldn't prove why!
Thanks in advance!
$W \setminus U$ cannot be an affine subspace indeed. Let us center it around a point $O \in W \setminus U$. $W$ then is a vector space with the point $O$ serving as the zero vector (since W) is affine. $U$ is an affine hyperplane in $W$ since its dimension is one less.
We can therefore describe it as $A+ker(\phi)$, with $\phi$ a linear form and $A$ a point in $U$. Now we want to create a line that intersects $U$ at $A$ and goes trough two points outside of $U$.
Let us prove, that the line $(AO)$ does the job. Since the ring has at least three elements, we chose some $x \in K$, such that $x$ is neither $0$, nor $1$. Then we take the point: $O+x \vec{OA}$ It is definitely on the line (OA). Now let us prove that it is not in $U$. We have that $(O+x \vec{OA})-A=(x-1)\vec{OA}$. Therefore $\phi((O+x \vec{OA})-A)=(x-1) \phi(\vec{OA})$ and neither $(x-1)$ nor $\phi(\vec{OA})$ are zero.