In this question c) ii)
Having found the matrix of T with respect to B, in order to find a basis C such that T with respect to C is diagonal, it is necessary to find the eigenvectors and eigenvalues for [T]b and use those to find [T]c as illustrated by this solution:
However, I am confused as to why [T]c is indeed diagonal. Taking the last vector in the basis C:
T(x+1) = T(1) + T(x) = (1+x+x^2) + (2x + x^2) = 1 + 3x + 2x^2 = 2(x^2-1) + 3(x+1)
Then surely the last column vector of [T]c should be (0,2,3) transposed. And this would mean [T]c is not diagonal.
What am I doing incorrectly here?
The book is wrong. The final eigenpolynomial of $T$ is $x+x^2$, not $1+x$. They did get the right vector $(0,1,1)^T$, but probably got the order of $B$ wrong when translating it back to a polynomial.