Linear Algebra Done Right · Problem 6C.2

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · orthogonal-complements

Problem

Suppose

U

is a subspace of

V

with basis

u_1, \ldots, u_m

and

u_1, \ldots, u_m, v_1, \ldots, v_n

is a basis of

V

. Prove that if the Gram–Schmidt procedure is applied to the basis of

V

above, producing a list

e_1, \ldots, e_m, f_1, \ldots, f_n

, then

e_1, \ldots, e_m

is an orthonormal basis of

U

and

f_1, \ldots, f_n

is an orthonormal basis of

U^\perp