Question

Characteristic polynomial of transformation question, help please!! :)

Answer 1

I have that \[\det(T-\lambda I) = \det(\left[\begin{matrix}\alpha-\lambda & 0\\ 0 & \beta-\lambda\end{matrix}\right] = (\alpha-\lambda)(\beta-\lambda) \] is the characteristic polynomial
But the solution gives \[(\alpha-\lambda)^2(\beta-\lambda)^3\] where 2 and 3 are dim U and dim W, why do these become multiplicites?

Answer 2

actually, your matrix representation of T is incorrect. it is given that U has dim 2 and W has 3. So, a basis of eigenvectors of V can be {u1,u2,w1,w2,w3} where{u1,u2} is the basis of U and {w1,w2,w3} is the basis of W. Now that you have a basis of eigenvectors, you can try constructing the matrix of T with this as the basis. The matrix will be 5x5. Try it.