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Adjoint transformation statement.

  1. Jun 7, 2014 #1
    I have doubts regarding a statement related to the following proposition: Let ##(V,<,>)## be a finite dimensional vector space equipped with an inner product and let ##f:V \to V## be a linear transformation, then the following statements are equivalent:

    1)##<f(v),f(w)>=<v,w>## for all ##v,w \in V##
    2) ##f^* \circ f=f \circ f^*=id_V##

    I've read and understood the proof of the equivalence between these two statements but I have a major doubt with the following: In the part 1) ##\implies## 2), in the textbook they prove 1) ##\implies f^* \circ f=id_V## and then affirm "As V is a finite dimensional vector space, ##f^* \circ f=id_V \implies f \circ f^*=id_V##. I don't understand why that implication is true, I would appreciate if someone could explain it to me.
     
  2. jcsd
  3. Jun 8, 2014 #2

    hilbert2

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    It's proved here in theorem 4, part 7), that in a finite-dimensional vector space the left and right inverses of a transformation are equal: http://www-math.mit.edu/~dav/onesidedCORR.pdf [Broken] .
     
    Last edited by a moderator: May 6, 2017
  4. Jun 8, 2014 #3

    Fredrik

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    The equality ##f^*\circ f=\mathrm{id}_V## implies that f* is surjective. A linear operator on a finite-dimensional vector space is surjective if and only if it's injective. (Can you prove those two statements?) So f* is bijective, and therefore invertible.
    \begin{align}
    &f^*\circ f =\mathrm{id}_V\\
    &\forall x~~ f^*(f(x)) =x\\
    &\forall x~~ (f^*)^{-1}(f^*(f(x)))= (f^*)^{-1}(x)\\
    &\forall x~~ f(x) = (f^*)^{-1}(x)\\
    &f =(f^*)^{-1}(x)
    \end{align} You could also just "apply ##(f^*)^{-1}## from the left" to both sides of the first equality, and then conclude that ##f=(f^*)^{-1}##, but that would be a bit suspicious IMO, because you would be using that ##\circ## is associative, and maybe what you have proved earlier is just that ##(f\circ g)\circ h=f\circ (g\circ h)## when f,g,h are all bijective. (We don't know that f is bijective at the start of the calculation. That's a conclusion we can make when we get to line 4).
     
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