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Column space of A'A?

  1. Jul 28, 2011 #1
    Let A be an n x p matrix with real entries and A' be its transpose. Is the column space of A'*A the same as the column space of A'. Obviously, the column space of A'*A is a subset of the column space of A' but can I show the other way? Thanks!
     
  2. jcsd
  3. Jul 29, 2011 #2
    Well, I figured it out if anyone is interested.

    Using the argument here (http://en.wikipedia.org/wiki/Rank_(linear_algebra)) under rank of a "Gram matrix" with real entries and the rank + nullity equals number of columns theorem you can show the rank of A equals the rank of A'*A.

    Thus, the rank(A')=rank(A)=rank(A'*A). So the column space of A' has the same dimension as the column space of A'*A and since the column space of A'*A is a subset of the column space of A' as vector spaces of the same dimension they are the same.

    I think that's right!
     
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