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

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SUMMARY

The column space of the matrix A'*A is equivalent to the column space of its transpose A' when A is an n x p matrix with real entries. This conclusion is drawn from the established fact that the rank of A equals the rank of A'*A, as supported by the rank-nullity theorem. Consequently, since both column spaces have the same dimension and one is a subset of the other, they are indeed identical.

PREREQUISITES
  • Understanding of matrix operations, specifically transposition and multiplication.
  • Familiarity with the concepts of column space and vector spaces.
  • Knowledge of the rank-nullity theorem in linear algebra.
  • Basic comprehension of Gram matrices and their properties.
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  • Study the properties of Gram matrices in linear algebra.
  • Learn about the rank-nullity theorem and its applications.
  • Explore the implications of matrix rank on linear transformations.
  • Investigate the relationship between column space and row space in matrices.
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Students and professionals in mathematics, particularly those studying linear algebra, as well as data scientists and engineers working with matrix computations and transformations.

MichaelL.
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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!
 
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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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