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FTIM question

  1. Mar 25, 2009 #1
    Sorry all my vectors look like superscripts, don't know what that's about.

    1. The problem statement, all variables and given/known data

    What is wrong with the following argument?

    Let A be an arbitrary m x n matrix. The vector A[tex]\vec{x}[/tex] is obviously in CA so it can't be in N(AT) unless it's the zero vector, since CA is orthogonal to N(AT). Thus the only solution to ATA[tex]\vec{x}[/tex]=[tex]\vec{0}[/tex] is [tex]\vec{x}[/tex]=[tex]\vec{0}[/tex] and ATA is an invertible matrix (by FTIM).

    2. Relevant equations

    The Fundamental Theorem of Invertible Matrices.

    3. The attempt at a solution

    I don't know if I understand the (incorrect) reasoning behind this argument. Mainly, I don't understand the connection between the end of the second sentence and the first half of the last sentence. What does CA and N (AT) being orthogonal have to do with the equation ATA[tex]\vec{x}[/tex]=[tex]\vec{0}[/tex]?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 25, 2009 #2

    HallsofIvy

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    It would help a lot if you would tell us (1) what CA and N(AT) mean, and (2) what this argument is supposed to prove.

     
  4. Mar 25, 2009 #3
    CA is the column space of the matrix A and N(AT) is the null space of A transpose. Sorry, I thought that was standard notation.
    The argument says that given an arbitrary matrix A, the matrix ATA (the matrix you get when you multiply A by A transpose on the left) is invertible. The point of the problem is to recognize that this isn't true and find the faulty reasoning.
     
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