To clarify why translation is not a linear transformation (even though it may seem like it). A linear transformation takes a weighted sum of its components. In this case x' = ax + cy. A translation can makes the coordinates (0,0) into (1,1). However there doesnt exist an a,c such that 1 = a0 + c0. S this is why translation is not linear
that's true, I didn't think about this! It's good that you pointed this out.