Linear Algebra Done Right · Problem 27

Polynomials — Sheldon Axler

Mathematics — Linear Algebra · null-space-range-dual-map

Problem

Suppose

T \in \mathcal{L}(\mathcal{P}_5(\mathbf{R}))

and

\operatorname{null} T' = \operatorname{span}(\varphi)

, where

\varphi

is the linear functional on

\mathcal{P}_5(\mathbf{R})

defined by

\varphi(p) = p(8)

. Prove that

\operatorname{range} T = \{p \in \mathcal{P}_5(\mathbf{R}) : p(8) = 0\}.