Linear Algebra Done Right · Problem 7A.5

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · adjoints

Problem

Suppose

T \in \mathcal{L}(V, W)

. Suppose

e_1, \ldots, e_n

is an orthonormal basis of

V

and

f_1, \ldots, f_m

is an orthonormal basis of

W

. Prove that

\|Te_1\|^2 + \cdots + \|Te_n\|^2 = \|T^* f_1\|^2 + \cdots + \|T^* f_m\|^2.

The numbers

\|Te_1\|^2, \ldots, \|Te_n\|^2

in the equation above depend on the orthonormal basis

e_1, \ldots, e_n

, but the right side of the equation does not depend on

e_1, \ldots, e_n

. Thus the equation above shows that the sum on the left side does not depend on which orthonormal basis

e_1, \ldots, e_n

is used.