Column Space of A'*A: Subset of A'?

[MichaelL.]

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!

 

[MichaelL.]

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!