16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 4.2.9

Polynomials

Mathematics — Linear Algebra · integration-differentiation-operators-polynomials

Problem

Let

F

be a subfield of the complex numbers and let

T

D

be the transformations on

F[x]

defined by

T\!\left(\sum_{i=0}^n c_i x^i\right) = \sum_{i=0}^n \frac{c_i}{1+i}\, x^{i+1}

and

D\!\left(\sum_{i=0}^n c_i x^i\right) = \sum_{i=1}^n i c_i x^{i-1}.

((a)) Show that

T

is a non-singular linear operator on

F[x]

. Show also that

T

is not invertible. ((b)) Show that

D

is a linear operator on

F[x]

and find its null space. ((c)) Show that

DT = I

, and

TD \neq I

. ((d)) Show that

T[(Tf)g] = (Tf)(Tg) - T[f(Tg)]

for all

f

g

F[x]

. ((e)) State and prove a rule for

D

similar to the one given for

T

in (d). ((f)) Suppose

V

is a non-zero subspace of

F[x]

such that

Tf

belongs to

V

for each

f

V

. Show that

V

is not finite-dimensional. ((g)) Suppose

V

is a finite-dimensional subspace of

F[x]

. Prove there is an integer

m \geq 0

such that

D^m f = 0

for each

f

V