16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 3.5.13

Linear Transformations

Mathematics — Linear Algebra · product-of-linear-functionals

Problem

Let

F

be a subfield of the field of complex numbers and let

V

be any vector space over

F

. Suppose that

f

and

g

are linear functionals on

V

such that the function

h

defined by

h(\alpha) = f(\alpha)g(\alpha)

is also a linear functional on

V

. Prove that either

f = 0

g = 0