In Borevich & Shafarevich's Number Theory, the authors define integral equivalence of quadratic forms as follows:
Two forms of the same degree with rational coefficients are called integrally equivalent if each can be obtained from the other by a linear change of variables with rational integer coefficients.
They further state a second definition
In the case of forms which depend on the same number of variables, this is equivalent to saying that one of the forms can be transformed into the other by a linear change of variables with unimodular matrix.
But I cannot see how to prove the equivalence of the two definitions. Specifically, it seems to me that to show the first definition implies the second, one has to prove that the two linear changes in the first definition are inverse of each other, or that both changes are nonsingular. Can anyone help?
edit: Mr. Stucky's example in the comments has shown that the two changes occurred in the first definition need not be invertible. But still, I think to prove that the first definition implies the second, one has to show the existence of two nonsingular changes, as is pointed out by Stucky.