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.
My attempt:
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?