Question

find a basis for R^3 relative to which the matrix for T is diagonal

Answer 1

\[ T =\left[\begin{matrix}0 & -1 & 1 \\ -1 &0 & 1 \\ 1&1&0 \end{matrix}\right]\]

Answer 2

For a transformation to be represented by a diagonal matrix means for the basis to be composed of the eigenvectors of the transformation. Thus, simply find the eigenvectors of \(T\), and the corresponding diagonal matrix will have the eigenvalues as entries.