Linear Algebra Done Right · Problem 11

Polynomials — Sheldon Axler

Mathematics — Linear Algebra · dual-basis

Problem

Suppose

v_1, \ldots, v_n

is a basis of

V

and

\varphi_1, \ldots, \varphi_n

is the corresponding dual basis of

V'

. Suppose

\psi \in V'

. Prove that

\psi = \psi(v_1)\varphi_1 + \cdots + \psi(v_n)\varphi_n.