Formatting messed up (I forgot about how the mark-up is handled).
Hopefully this works.

> So, x1 and x2 both are associated [in the backward direction from/wrt
the previous usage of "associated with"] ...

I was just bracketing out commentary information, indicating that the
present usage of the word "associated" was essentially a semantic inverse
of the previous usage of the same word. I was trying not to disrupt the
flow of the explanation too much by this interjection of information, as
well as distinguishing one type of parenthetical with this (more meta)
usage of an aside. So, the hyperlinked text should be considered merely
an aside.

> x3 = Transpose[(r,0)],+,* for real r

Should be: x3 is the vectorspace of column vectors of two entries, the
first entry being any real number and the second entry being 0
identically. (Basically any vector pointing along a given line.)

> [the vectorspace of (column) vectors of form (r,0) for real r]

Same thing. I was just trying to explain what I meant the first time, but
the explanation was hyperlinked.


> will tell us that any transformation that can work in x2 will be of
form:
> a b 0 b
> =
> c d 0 d.


I was making a square matrix of four entries (id est a 2-by-2 matrix).
The entry in the first column and first row is a = 0, the entry in the
second column and first row is b, the entry in the second row and first
column is c = 0, and the entry in the second row and second column is d.