16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 8.4.6

Inner Product Spaces

Mathematics — Linear Algebra · unitary-operators-reflection

Problem

Let

V

be a finite-dimensional inner product space, and let

W

be a subspace of

V

. Then

V = W \oplus W^\perp

, that is, each

\alpha

V

is uniquely expressible in the form

\alpha = \beta + \gamma

, with

\beta

W

and

\gamma

W^\perp

. Define a linear operator

U

U\alpha = \beta - \gamma

. (a) Prove that

U

is both self-adjoint and unitary. (b) If

V

R^3

with the standard inner product and

W

is the subspace spanned by

(1, 0, 1)

, find the matrix of

U

in the standard ordered basis.