I have a question about orthogonal transformations.
If $T$ is an orthogonal transformation from $V$ to $V$, should the representation matrix with respect to any orthonormal basis of any inner product be an orthogonal matrix?
I'll be more specific, I was given $Tu=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}u$, $T: \mathbb R^{2} -> \mathbb R^{2}$ and I was asked is there an inner product where this transformation is orthogonal.
What I want to say is, since the standard basis is orthonormal with respect to the standard inner product (dot product), and the matrix representation of $T$ with respect to that basis is not orthogonal, then the matrix representation of $T$ with respect to ANY orthonormal basis (not necesarily with respect to the regular inner product) is not orthogonal.
Is it correct to say such a thing?