- #1
neomasterc
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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 Tt(f)=cf.
Tt denotes T transpose.
Homework Equations
Tt(f)=f°T. Ev(f)=f(v).
Let V*=HomF(V,F)
V*=(V*)*
E:V→V** i.e. E(v)=Ev
Theorem: E is an isomorphism iff V is finite dimensional
The Attempt at a Solution
Ev(Tt(f))=f°T(v)=cf(v).
But now I need to show there is f such that Tt(f)=cf, not just for specific v that satisfies T(v)=cv.