What does n mean? Does it mean a vector normal to v?
The transpose equaling the inverse means that the columns of A form an orthonormal basis.
i think it's a typo. i believe the OP is asking to show that every inner-product preserving linear operator has an orthogonal matrix in the standard orthonormal basis, and that an orthogonal matrix represents an inner-product preserving linear operator.
the whole of this proof rests on the fact that (for the standard inner product in R^{n}):
so if:
for all u, v, then:
that is:
for all u, so
so that:
for all v, thus:
these steps are all reversible.