The task is to find any affine transformation that will swap the following two lines: $$L_1:(1,1,1) + span((1,0,2))$$ $$L_2:(1,0,1) + span((1,0,-1))$$ From what I understand there is a number of equations I can make: $$f((1,1,1))=(1,0,1)$$ $$f((1,0,1))=(1,1,1)$$ I am not sure how to create the other two equations that I need. I also understand that an affine transformation consists of a linear transformation and a translation. Does this translation have to be $(1,1,1) - (1,0,1)$ or what else could it be.
I'm really confused by the affine transformations.
I think the key observation is that an affine transformation will take lines to lines. As a line is determined by two points, the image of a line is determined by the images of any two points on it. This will yield 4 equations that determine a 3x3 matrix (the linear transformation) and a translation. You may find the computations easier if you first translate the lines so that one of them passes through the origin.