Chris

@ArturoMagidin This was part of exercise 5.C #5 (a direction of the if and only if). My professor solved it through induction on the dimension of V. If my notes are right, it looks like he showed that for any vector space V' with dim(V') < dim(V) and T' in L(V) such that V' = null(T'-lambdaI) + (direct) range(T'-lambdaI) for all lambda in C, T' is diagonalizable. Then he showed that if T' is diagonalizable, then T is diagonalizable. Not sure how to use the math symbols in chat.

@ArturoMagidin No worries if you can't. I can try to discuss it with my professor this week as well.

Just updated the question, sorry

@ArturoMagidin My text doesn't go into generalized eigenvectors until much later, and my book solution uses answers to previous questions to solve this, which I don't think is an appropriate way for me to answer this. I was thinking there must be some way to use theorems of direct sums or eigenspaces to solve this, but haven't made progress.

@angryavian Yes, this is for all $\lambda \in \mathbb{C}$.