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Proof of invertibility

  1. Feb 4, 2016 #1
    I am trying to understand the following basic proposition about invertibility: a linear map is invertible if and only if it is injective and surjective.
    Now suppose ##T## is a linear map ##T:V\rightarrow W##. The book I read goes the following way in proving the proposition in the direction when the surjectivity and injectivity act as the condition. Suppose ##T## is injective and surjective and a vector ##w \in W##. Then define ##Sw## to be the unique element of V such that ##TSw = w##. Therefore ##TS## is an identity transformation in ##W##.
    Now, I understand that ##Sw## is unique because ##T## is injective, but I don't know how the surjectivity contributes to guarantee that ##Sw## which satisfies ##TSw = w## does exist.
     
  2. jcsd
  3. Feb 4, 2016 #2

    Samy_A

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    ##T## being surjective implies that there is a ##v \in V## satisfying ##Tv=w##. That ##v## is ##Sw##.
    If ##T## is not surjective you can't be sure that there will be a ##Sw## satisfying ##TSw = w##.
     
  4. Feb 5, 2016 #3
    Sorry I am not getting your explanation. If ##T## is surjective then ##\textrm{range}(T) = W##, isn't it. How can this information be used to conclude that there is ##v## in the domain space ##V## which satisfies ##Tv=w##, while the surjectivity of ##T## concerns the range space ##W## not the domain space ##V##?
     
  5. Feb 5, 2016 #4

    Samy_A

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    Correct, surjectivity means ##T(V)=W##, that every element of ##W## lies in the range of ##T##.

    That means that for every ##w \in W## there is a ##v \in V## satisfying ##Tv=w##.
     
    Last edited: Feb 5, 2016
  6. Feb 6, 2016 #5

    HallsofIvy

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    "Surjective" says that "for any w in W, there exist at least one v in V such that T(v)= w. "Injective says that there is not more than one such v. If T is both "surjective" and "injective" then there exist exactly one such v and that is, by definition, T-1(w).
     
  7. Feb 6, 2016 #6
    Hi HallsofIvy, thank you.
    My background is actually physics and I am just near the beginning of thinking in an abstract manner. In this state of mine, rearrangement and choice of words to translate abstract mathematical line of reasoning really helps me.
     
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