1. ## Diagonalizablilty!! Urgent!! (2)

Let T be an invertible linear operator on a finite-dimensional vector space V.

a) Recall that for any eigenvalue λ of T, λ^-1 is an eigenvalue of T^-1. Prove that the eigenspace of T corresponding to λ is the same as the eigenspace of T^-1 corresponding to λ^-1.

b) Prove that if T is diagonalizable, then T^-1 is diagonalizable

2. If $\lambda$ is an eigenvalue of A, then $Ax= \lambda x$ for any x in the eigenspace corresponding to $\lambda$

Now apply $T^{-1}$ to both sides of that equation.

(b) is trivial if you think about what "diagonaizable" means.