[ReleaseCandidat]

> For example there is no linear map which maps a line (or segment) through the origin to a parallel line (or segment of the same length) that doesn't pass through the origin, even though these are clearly just the same object shifted around a bit.

That's why god (well, projective geometrists) made homogeneous coordinates. Without them geometry isn't much fun when using linear algebra (as you have way too many special cases).