Linear Algebra Done Right · Problem 19

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · inner-products

Problem

Suppose

v_1, \ldots, v_n

is a basis of

V

. Prove that there exists a basis

u_1, \ldots, u_n

V

such that

\langle v_j, u_k \rangle = \begin{cases} 0 & \text{if } j \neq k, \\ 1 & \text{if } j = k. \end{cases}