Linear Algebra Done Right · Problem 1

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · orthonormal-bases

Problem

Suppose

e_1, \ldots, e_m

is a list of vectors in

V

such that

\|a_1 e_1 + \cdots + a_m e_m\|^2 = |a_1|^2 + \cdots + |a_m|^2

for all

a_1, \ldots, a_m \in \mathbf{F}

. Show that

e_1, \ldots, e_m

is an orthonormal list. This exercise provides a converse to 6.24.