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Math Help - Flaw in proof

  1. #1
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    Flaw in proof

    I need to find the flaw in the following proof. Can anyone help me?

    Suppose A has a right inverse (say B)
    AB = I
    A^(T)AB = A^(T)
    B = (A^(T)A)^(-1) (A^(T))
    BA = (A^(T)A)^(-1) (A^(T)A) = I
    Therefore B is also a left inverse of A
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  2. #2
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    Re: Flaw in proof

    Quote Originally Posted by page929 View Post
    I need to find the flaw in the following proof. Can anyone help me?

    Suppose A has a right inverse (say B)
    AB = I
    A^(T)AB = A^(T)
    B = (A^(T)A)^(-1) (A^(T))
    BA = (A^(T)A)^(-1) (A^(T)A) = I
    Therefore B is also a left inverse of A
    That argument assumes that A^{\textsc t}\!A is invertible. In the case where A and B are matrices, you can check (by considering determinants) that A and A^{\textsc t} (and hence their product) are indeed invertible. But in some other situations, for example if A and B are linear operators on an infinite-dimensional space, the result fails, and you can have operators which have a right inverse but not a left inverse.
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