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Nullspace of matrix A and the nullspace of A^T*A

  1. Nov 20, 2012 #1
    1. The problem statement, all variables and given/known data
    A is matrix m*n
    show that nullspace of A is the subset of nullspace of A^T*A


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 20, 2012 #2

    HallsofIvy

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    Have you tried anything at all? The standard way to prove "X is a subset of Y" is start "If x is in X" and use the properties of X and Y to conclude "then x is in Y".

    So suppose x is in the null space of A. What can you say about [itex]A^T Ax[/itex]?
     
  4. Nov 20, 2012 #3
    I know Ax=0 and i guess A^T*A is also 0
     
  5. Nov 20, 2012 #4

    haruspex

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    You mean, you guess that AT*Ax is also 0, right? But why do you need to guess? If Ax = 0, you can easily prove AT*Ax = 0.
     
  6. Nov 20, 2012 #5
    Done prove ?
    So x is also in the nullspace of A^TA so done ? serioulsy ...
     
    Last edited: Nov 20, 2012
  7. Nov 20, 2012 #6
    The set of all values of x such that Ax=0 is the Null space of A.
    The set of all values of x such that (A^(T)*A)x=0 is the Null space of A^(T)*A
    Since A^(T)*A is linear, (A^(T)*A)x = A^(T)*(Ax)
    Therefore A^(T)*(Ax)=0
    A^(T)*(0) = 0 - For all values of x such that Ax = 0 - Therefore the set of solutions to Ax=0 is a subset of A^(T)*A

    Now just turn that into mathematical notation.
     
    Last edited: Nov 20, 2012
  8. Nov 20, 2012 #7
    i love you guys thanks
     
  9. Nov 20, 2012 #8
    Yeah, all the babes dig the math nerds ;)
     
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