16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 10.2.11

Bilinear Forms

Mathematics — Linear Algebra · adjoint-operator-symmetric-form

Problem

Let

V

be a finite-dimensional vector space and

f

a non-degenerate symmetric bilinear form on

V

. Show that for each linear operator

T

V

there is a unique linear operator

T'

V

such that

f(T\alpha, \beta) = f(\alpha, T'\beta)

for all

\alpha, \beta

V

. Also show that

(T_1 T_2)' = T_2' T_1'

(c_1 T_1 + c_2 T_2)' = c_1 T_1' + c_2 T_2'

(T')' = T.

How much of the above is valid without the assumption that

T

is non-degenerate?