In quantum mechanics, in the context of symmetry transformations, it is often said that for a transformation $T$ to conserve probabilities it must be unitary.
But by performing any (even non-unitary) transformation on the system we are just taking account for the fact we are looking at it in a different way (i.e. using different basis vectors).
By looking at the system in a different way I cannot see how we could change the probabilities of a measurement (as long as these probability where calculated correctly, which may need the introduction of a matrix into the scalar product).
Am I correct? If so why is it said that only unitary matrices conserve probabilities and if not why not?