16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 4.3.4

Polynomials

Mathematics — Linear Algebra · lagrange-polynomials-linear-operator

Problem

Let

p = (x-2)(x-3)(x-1)

and let

T

be any linear operator on

R^4

such that

p(T) = 0

. Let

P_1, P_2, P_3

be the Lagrange polynomials of Exercise 3, and let

E_i = P_i(T)

i = 1, 2, 3

. Prove that

E_1 + E_2 + E_3 = I, \qquad E_i E_j = 0 \text{ if } i \neq j,

E_i^2 = E_i, \quad \text{and} \quad T = 2E_1 + 3E_2 + E_3.