Let u and v be vectors in Rn, and let T be a linear operator on Rn. Prove that T(n) * T(v) = n * v if and only if AT = 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 Rn):
for all u, v, then:
for all u, so
for all v, thus:
these steps are all reversible.