Let P3 be the real vector space of polynomials in the variable x of degree at most 3 with real

Question

Let P3 be the real vector space of polynomials in the variable x of degree at most 3 with real

coefficients, and let T : P3 → P3 be the endomorphism given by T (p(x)) := (d/dx)[(x2 − x)p(x)]

(a) Using the ordered basis {1, x, x2, x3} for P3 , find the matrix representing T .

(b) Suppose q(x) ∈ P3 is arbitrary. Give a very brief reason why there is a polynomial p(x) ∈ P3 satisfying T (p(x)) = q(x) .

Let P3 be the real vector space of polynomials in the variable x of degree at most 3 with real
coefficients, and let T : P3 – P3 be the endomorphism given by T(p(x)) :=
d
dx
(2 2 – 20) p(z) .
(a) Using the ordered basis {1, x, x2, x3} for P3, find the matrix representing T.
(b) Suppose q(x) E P3 is arbitrary. Give a very brief reason why there is a polynomial
p(x) EP3 satisfying T(p(x) ) = q(x) .
Math