Take a multi-linear function(or functional) M that takes m arguments V1…Vm, each with a dimension n. Consider only the case where m=n. Let there be a change of basis performed on the arguments(V1...Vm) by the transformation matrix T. The affect on the output of M is one dimensional and can be characterized by the determinate of T. Thus, the effect of the output of M from the change of basis of the arguments is purely multiplication of a constant. Is this correct?

Or, is the determinant of T only explaining the effect of T with respect to the canonical basis of which the determinant is equal to 1?