Linear Algebra Done Right · Problem 14

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · gram-schmidt-procedure

Problem

Suppose

V

is a real inner product space and

v_1, \ldots, v_m

is a linearly independent list of vectors in

V

. Prove that there exist exactly

2^m

orthonormal lists

e_1, \ldots, e_m

of vectors in

V

such that

\operatorname{span}(v_1, \ldots, v_k) = \operatorname{span}(e_1, \ldots, e_k)

for all

k \in \{1, \ldots, m\}