Show that two lines in an affine space are parallel iff they have the same direction.
I don't know how to proceed. We know that any subspace of an affine space $A$ based on a vector space $V$ is of the form $O+W=\{\ O+w |w\in W\}\ $, for some subspace $W$ of $V$. (Here, the plus between $O$ and $w$, denotes the action of $V$ on $A$.), and $W$ is called the direction of the affine subspace. Now if $PQ$ and $RS$ are two lines, they are subspaces too, So, $PQ= O+W$ and $RS=O^{'}+W^{'}$. What I suppose is that we need to prove $W=W^{'}$ iff $PQ$ is parallel to $RS$. But I can't proceed after that
Thanks in advance!!
You need to show three things, and the special case of identical lines is worth considering for each of them.
If they have the same direction, they lie in a plane. If you define a plane as $O + W_1 + W_2$, i.e. a starting point and two directions, then you can pick $W_1=W$ and $W_2=\{\lambda(O-O')\mid\lambda\in\mathbb R\}$, i.e. one direction vector equal to that of the line and the other equal to the difference between the starting points of the lines. If those two directions are the same, the lines coincide and you can either pick a second direction arbitrarily or argue that a single line is alwas coplanar with itself.
If they have the same direction they have no point in common. More precisely, if they have the same direction and one point in common, then they have all points in common. So you have $O + w = O' + w'$ for some $w,w'\in W$. How you continue from here depends a bit on how you define a one-dimensional linear subspace, but if it has a basis, you'd express $w$ and $w'$ with respect to that and then show that there is a formula for the coefficients which maps points from one line to those of the other.
If they are coplanar and have no points in common, then they have the same direction. If you know that your plane has a basis, you can express the bases of the line directions with respect to that. Then you can assume different directions and show that you can compute a point of intersection.