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Homework Help: Dual spaces-Existence of linear functional

  1. Nov 6, 2011 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant 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


    3. 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.
     
  2. jcsd
  3. Nov 6, 2011 #2
    How would you define f?? Obviously you will want to have f(v)=v. How would you define it for other vectors?? Maybe set f(w)=0 for some other vectors??
     
  4. Nov 6, 2011 #3
    f is just a linear transformation in V*=HomF(V,F), so you cannot have f(v)=v, f:V→F.
     
  5. Nov 6, 2011 #4
    f(v)=1 I meant.
     
  6. Nov 6, 2011 #5
    We cannot define f(kv)=1 and f(w)=0 for w≠v, since if a is a multiple of v and b is a multiple of v, a+b is a multiple of v, so 1=f(a+b)≠f(a)+f(b)=2. So f wont be a linear transformation.
     
  7. Nov 6, 2011 #6
    Indeed we can't. Can you solve that?? Don't put ALL the vectors equal to 0, but only some. Do you see it??
     
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