Question

Let T be a linear endomorphism on a vector space V over a field F with n = Pr(t) the minimal polynomial of T. Problem 1. Show that T is invertible if and only if PT (0) # 0. dim(V) 1. We denote by

Answer 1

Let T be a linear endomorphism on a vector space V over a field F with n = Pr(t) the minimal polynomial of T. Problem 1. Show that T is invertible if and only if PT (0) # 0.The following are the given conditions:Let PT (x) be the characteristic polynomial of T and n = Pr(t) be the minimal polynomial of T. And it's known that deg(PT (x)) = dim(V).Suppose T is invertible. Then we can assume PT (T) = 0 implies the zero endomorphism. Thus, T is not a root of the characteristic polynomial. Hence, PT (T) # 0.Therefore, PT (0) = (-1)^n det(T) # 0 as T is invertible. This shows that PT (0) is nonzero.Suppose PT (0) is nonzero. Since PT (x) is the characteristic polynomial, we see that T is diagonalizable. Thus, the only possible roots of the minimal polynomial are 0's. Since PT (0) is nonzero, T is not a root of the characteristic polynomial. Thus, PT (T) = 0 implies the zero endomorphism. Therefore, T is invertible.This shows that T is invertible if and only if PT (0) # 0. Answer: PT (0) # 0.