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Proving subset of a Nullspace

  1. Apr 29, 2014 #1
    1. The problem statement, all variables and given/known data

    Given matrix A (size m x n), prove N(A) is subset of N( A^t A).

    A^t is matrix A transposed.

    2. Relevant equations



    3. The attempt at a solution

    My assumption is m < n, using definition of nullspace, I ended up with N( A^t A) = a set of zero vector, while N(A) is not entirely included in N( A^t A).

    Thank You.
     
  2. jcsd
  3. Apr 29, 2014 #2

    jbunniii

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    If ##Ax = 0##, then what is ##A^t A x##?
     
  4. Apr 29, 2014 #3
    A^t A x= 0

    and how do I justify the subset part?
     
  5. Apr 29, 2014 #4

    jbunniii

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    If ##X## and ##Y## are sets, how do you prove that ##X \subset Y## in general?
     
  6. Apr 29, 2014 #5
    I need to show that elements in X also belongs to Y.
     
  7. Apr 29, 2014 #6

    jbunniii

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    Yes, that's right. So what is the defining property of an element of ##N(A)##? In other words, ##x \in N(A)## if and only if ...?
     
  8. Apr 29, 2014 #7
    x ε N(A) iff Ax = 0 and since A^t A x = 0 then x ε N(A^t A).
    Therefore x belongs to both N(A) and N(A^t A).

    Is this correct?
     
  9. Apr 29, 2014 #8

    jbunniii

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    You have the right idea, but you need to state the logic correctly. The goal is to prove that if ##x \in N(A)## then ##x \in N(A^t A)##.

    So, suppose ##x \in N(A)##. Then by definition, ##Ax = 0##. Therefore...?
     
  10. Apr 29, 2014 #9

    LCKurtz

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    Do you understand why that is zero? You stated it but didn't prove it.
     
  11. Apr 29, 2014 #10
    Ok I think I got it.

    to prove that if ##x \in N(A)## then ##x \in N(A^t A)##.

    ##x \in N(A)##. By definition, ##Ax = 0##
    therefore ##A^t A x = 0## which means ##x \in N(A^t A)## as well.

    Therefore ##N(A) \subset N(A^t A)##.

    Thanks a lot jbunniii.
     
    Last edited: Apr 29, 2014
  12. Apr 29, 2014 #11

    jbunniii

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    Looks good!
     
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