The Wikipedia page on orthogonal transformation says:
In linear algebra, an orthogonal transformation is a linear transformation T : V → V on a real inner product space V, that preserves the inner product. That is, for each pair u, v of elements of V, we have
$$\langle u,v\rangle =\langle Tu,Tv\rangle$$
Just given this definition, is it possible to prove that orthogonal transformations can only have $\pm 1$ determinant? I know the proof for orthogonal matrices but not for this general definition of orthogonal transformation. Or could someone tell me how to show that the matrix representation (with respect to an orthonormal basis) of an orthogonal transformation is an orthogonal matrix?