16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 3.7.8

Linear Transformations

Mathematics — Linear Algebra · trace-linear-functional-isomorphism

Problem

Let

V

be the vector space of

n \times n

matrices over the field

F

. (a) If

B

is a fixed

n \times n

matrix, define a function

f_B

V

f_B(A) = \operatorname{trace}(B^t A)

. Show that

f_B

is a linear functional on

V

. (b) Show that every linear functional on

V

is of the above form, i.e., is

f_B

for some

B

. (c) Show that

B \to f_B

is an isomorphism of

V

onto

V^*