Kunze Linear Algebra · Problem 3.7.8

Linear Transformations

Mathematics — Methods · linear-functionals-via-trace

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^*

Intuitive explanation

The trace functional $\operatorname{trace}(B^t A)$ is a clever way to extract a single scalar from a matrix product—it's really just a dot product in disguise. Part (a) asks you to verify linearity by checking that $f_B$ respects addition and scalar multiplication of matrices. For part (b), the key insight is that any linear functional must assign values to the standard basis matrices $E_{ij}$ in a way that's completely determined by those choices; you'll reconstruct the matrix $B$ from those values. Part (c) then shows this correspondence is bijective and structure-preserving, establishing that the dual space $V^*$ is isomorphic to $V$ itself—a beautiful symmetry that hinges on the trace's ability to encode all linear information about matrices.

Intuitive explanation

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