1. The problem statement, all variables and given/known data From Serge Lang's "Linear Algebra, 3rd Edition", pg 51 exercise 9. Prove that the image is equal to a certain set S by proving that the image is contained in S, and also that every element of S in in the image. 9. Let F:R2→R2 be the mapping defined by F(x,y)=(xy,y). Describe the image under F of the straight line x=2. 2. Relevant equations 3. The attempt at a solution I first simply drew out a vertical line of x=2 and applied F to (2,y) for various values of y. F(2,0)=(0,0) F(2,1)=(2,1) F(2,2)=(4,2). I noticed it seemed to form a line which could be described by y=(1/2)x . I got a little stuck and looked at the answer and to my surprise the image was a line with slope 2. Below is the answer, and I just don't understand it... SOLUTION The image of F is the line whose equation is y=2x. Indeed, if (2,y) belongs to the line x=2, then F(2,y)=(2y,y), and clearly (2y,y) belongs to the line y=2x. Conversely, suppose v=2u; then F(2,v/2)=(v,v/2)=(v,u). I don't see why "clearly (2y,y) belongs to the line y=2x". If x is twice y in that tuple, wouldn't the equation be x=2y, or equivalently y=x/2 ? The other part I partly understand. I see why F(2,v/2)=(v,u) after applying the transformation, but I don't understand the significance of using v and u and setting up the equation v=2u.