Solved: Label the following statements as true or false. Assume that all vector spaces

Chapter 7, Problem 1

(choose chapter or problem)

Label the following statements as true or false. Assume that all vector spaces are finite-dimensional. (a) Every linear operator T has a polynomial p(t) of largest degree for which 7->(T) = TQ. (b) Every linear operator has a unique minimal polynomial. (c) The characteristic polynomial of a linear operator divides the minimal polynomial of that operator. (d) The minimal and the characteristic polynomials of any diagonalizable operator are equal. (e) Let T be a linear operator on an n-dimensional vector space V, p(t) be the minimal polynomial of T, and f(t) be the characteristic polynomial of T. Suppose that f(t) splits. Then f(t) divides \p(t)\n - (f) The minimal polynomial of a linear operator always has the same degree as the characteristic polynomial of the operator. (g) A linear operator is diagonalizable if its minimal polynomial splits. (h) Let T be a linear operator on a vector space V such that V is a T-cyclic subspace of itself. Then the degree of the minimal polynomial of T equals dim(V). (i) Let T be a linear operator on a vector space V such that T has n distinct eigenvalues, where n dim(V). Then the degree of the minimal polynomial of T equals n.