Problem: Determine whether the given map is an isomorphism of the first binary structure with the second.
with where is the determinant of matrix A.
I know that I need to complete four steps to prove isomorphism.
1) Define the function that gives isomorphism S to S'.
2) Show that is one-to-one.
3) " " is onto S'.
4) Show that for all .
For step one,
det(x) for all is in , so .
For step two,
If , det(x) = det(y). This is false, since determinants are not unique. det(x) can equal det(y) while x is not equal to y. Therefore, this map is not isometric since it is not one-to-one.
Then I stop there since it's not one-to-one, there's no need to prove the rest. But am I right? And if so, how would I prove that it is or isn't onto?