Let u and v be vectors in R^{n}, and let T be a linear operator on R^{n}. Prove that T(n) * T(v) = n * v if and only if A^{T} = A^{-1} where A is the standard matrix for T.
Any help would be great.
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.