I have trouble in verifying example 7.30 as the following in Linear Algebra Done Right (Third Edition).
7.30 Example: Consider the self-adjoint operator $T$ on $R^3$ whose matrix(with respect to the standard basis) is
$$ \left(\begin{array}{cc} 14 & -13 & 8 \\ -13 & 14 & 8 \\ 8 & 8 & -7 \end{array} \right). $$
As you can verify, $$ \frac{ \left( 1, -1, 0 \right) }{ \sqrt 2 }, \frac{ \left( 1, 1, 1 \right) }{ \sqrt 3 }, \frac{ \left( 1, 1, -2 \right) }{ \sqrt 6 } $$ is an orthonormal basis of $R^3$ consisting of eigenvectors of $T$, and with respect to this basis, the matrix of $T$ is the diagonal matrix
$$ \left(\begin{array}{cc} 27 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & -15 \end{array} \right). $$
I'm new in linear algebra, and I guess it might be very simple. But I still don't know how to verify it. Maybe my fundamental problem is, I don't know, how will the matrix of the linear map be changed with an basis to another basis?