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A'A and A have the same null space

  1. Oct 28, 2012 #1
    I'm trying to prove that the null space of A'A is the null space of A, this is what I have so far,

    1) A'Ax=0, non trivial solutions are a basis for the null space of A'A

    2) x'A'Ax=0

    3) (Ax)'Ax=0

    4) Since (Ax)'A is a linear combination of the col's of A, we see that the null space of A can be written as a linear combination of the basis for the null space of A'A.

    Therefore, they have the same null space.

    --> Is this proof valid? I am unsure if argument 4 holds ground, but it seems to make sense to me =P
     
  2. jcsd
  3. Oct 29, 2012 #2

    Fredrik

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    Is A' your notation for the transpose of A?
     
  4. Oct 29, 2012 #3
    It is, sorry for not making that clear!
     
  5. Oct 31, 2012 #4
    didnt think throguh your 4), but this is what you can do once you get to 3):

    Ax= 0 => (Ax)'Ax = (Ax|Ax) = ||Ax||^2 = 0 <=> Ax = 0

    where (,|,) denotes the scalar product and ||.|| is the norm induced by the scalar product ( ||u|| = sqrt(u|u) ).
     
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