Prove that the mapping F: P2(R) to P2(R) given by F(x1,x2,x3) = (x1x2, x2, x3) is not well-defined.
I know that to determine whether a mapping is well-defined, you should pick two points that are the same in P2(R) and show that the mapping transforms them the same way. So, would the points (1,2,3) and (2,4,6), which are scalar multiples of each other so are the same point in P2(R), an example showing why it is not well-defined? Because (1,2,3) is mapped to (2,2,3) and (2,4,6) is mapped to (8,4,6) and (2,2,3) and (8,4,6) are not scalar multiples of each other. Any help is appreciated.