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## Homework Statement

Let V be a finite dimensional vector field over F. Let T:V→V

Let c be a scalar and suppose there is v in V such that T(v)=cv, then show there exists a non-zero linear functional f on V such that T

^{t}(f)=cf.

T

^{t}denotes T transpose.

## Homework Equations

T

^{t}(f)=f°T. E

_{v}(f)=f(v).

Let V*=Hom

_{F}(V,F)

V*=(V*)*

E:V→V** i.e. E(v)=E

_{v}

Theorem: E is an isomorphism iff V is finite dimensional

## The Attempt at a Solution

E

_{v}(T

^{t}(f))=f°T(v)=cf(v).

But now I need to show there is f such that T

^{t}(f)=cf, not just for specific v that satisfies T(v)=cv.