16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 8.1.13

Inner Product Spaces

Mathematics — Linear Algebra · conjugation

Problem

Let

V

be a complex vector space. A function

J

from

V

into

V

is called a conjugation if

J(\alpha + \beta) = J(\alpha) + J(\beta)

J(c\alpha) = \bar{c}J(\alpha)

, and

J(J(\alpha)) = \alpha

, for all scalars

c

and all

\alpha

\beta

V

. If

J

is a conjugation show that: (a) The set

W

of all

\alpha

V

such that

J\alpha = \alpha

is a vector space over

R

with respect to the operations defined in

V

. (b) For each

\alpha

V

there exist unique vectors

\beta

\gamma

W

such that

\alpha = \beta + i\gamma