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Linear Transpose Help

  1. Sep 23, 2009 #1
    1. The problem statement, all variables and given/known data
    I have an idea on how to part 1, but I have no clue on how to do part 2 and 3.

    1.Suppose A is invertible. Check that (A-1)TAT=I and AT(A-1)T=I, and deduce that AT is likewise invertible with inverse (A-1)T.

    2. Suppose A is an mxn matrix with rank 1. Prove that there are nonzero vectors u element in Rm and v element in Rn such that A=uvT.

    3.Suppose A is an mxn matrix and x is an element of Rn satisfies (ATA)x=0. Prove that AX=0.


    2. Relevant equations

    For part one I'm guessing (AB)T=BTAT and (A-1A-1)=I

    3. The attempt at a solution
    Part 1. I know that some kind of way that it is due to the relationship of (A-1A-1)=I
     
  2. jcsd
  3. Sep 23, 2009 #2

    lanedance

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    (A-1A-1) does not equal I


    (A-1.A) = (A.A-1) = I

    what is

    (A-1.A)T ?
     
  4. Sep 23, 2009 #3
    (A.A-1) = I Im sorry I typed that wrong
     
  5. Sep 23, 2009 #4

    lanedance

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    ok so ideas for 1) ?

    for 2), what does it mean to be a matrix of rank 1? and uvT looks like an outer product, do you know how this is defined? Thinking about row reduction may help make the connection...

    for 3) think about multiplying both sides of your equation by something...
     
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