Converting from 2D coordinate Systems

[quasi426]

I am trying to get from one 2D coordinate system to another 2D coordinate system. I found 2 corresponding points and each system. I did the following:

[a1 a2 ] * [ x1,y1 ] = [u1,v1]
[a3 a4] [ x2,y2] [u2,v2]

or A*x = u

if I use matlab to solve A = u*inv(x)

when will A be a valid transformation matrix?
Are there other methods to find transformation matrices?

[quasi426]

This is not a homework question, i'm just doing some image processing and just need some help.