Btw, one should try to be logical. You wrote:

"Suppose that in the frame of the Earth, the following
sequence of events happen: 

e1: A rocket passes by the Earth at velocity v (as
     measured in the Earth frame).  The Earth-coordinates 
     for this event are x1=0, t1=0.

Now let's look at the same events from the point of view of
the rocket:
 
     e1: The Earth passes the rocket at velocity -v, as
     measured in the rocket frame. The rocket-coordinates
     for this event are x1' = 0, t1' = 0.

So our transformation equations are: 
(assuming that
       x' = A x + B t
       t' = C x + D t)

(1)    x' = A (x-vt)
(2)    t' = A (t-vx/c^2)"

According to your premises, when x1 = 0 and t1 = 0, x1' = 0 and t1' = 0,
or vice versa.
Or, from relation (2), i.e. t1' = A (t1-vx1/c^2), one also gets t1' = 0
when t1 = vx1/c^2. This contradicts you premises, and proves that your
transformation equations cannot be linear.

Don't be a moron like me (cf. feuerbac@thphys.uni-heidelberg.de)!

Marcel Luttgens