I don't understand 2 sub questions from a larger question(they don't have anything in common though) i'll write them and then show what i think/did:
a)$A \in M_{nxn}^F$ matrix that cannot be diagonalized. can there exist a polynom $p(t) \in F[t]$ of degree less than n so that $[p(A)]^2=0$ when 1)$F=C$? 2)when $F=R$?
b)(a claim) ${v_1,...,v_n}$ base of V and $T: V \to V$ follows:$Tv_1=0$, $T_{vi}=v_{i-1} $ ($2 \leq i \leq n$), so there exists $T^k=0$ in $1 \leq k < n$
what i think:
a)when i saw this question i was thinking about rotation matrices as they're only diagonizable over C and not R, but when i saw that the question refersto an annihilating(zeroing) polynom of degree less than n, which is incorect because for a matrix to be diagonizable, it has to have n linearily independent eigen vectors(rank n), which means that according to the question details it is not possible not under C nor R, so both are false claims.
b)i don't know how to solve it (check if the claim is correct or false) and would appreciate an explanation if possible. knowing that $T_{v2}=v_1$ and $t_{v1}=0$, how can i use this and the basis information (${v_1,...,v_n}$ basis of V) to check if the claim is correct(if there exists a $T^k=0$ in $1 \leq k < n$)?
thank you very much for your help. really interested in learning how to solve b).