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Math Help - Rank and nonsingularity

  1. #1
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    Rank and nonsingularity

    Let A be an {m\times n} matrix
    B be an {m\times m} nonsingular matrix and
    C be an {n\times n} nonsingular matrix

    Show that rank(A)=rank(BA)=rank(AC)
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  2. #2
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    Quote Originally Posted by altave86 View Post
    Let A be an {m\times n} matrix
    B be an {m\times m} nonsingular matrix and
    C be an {n\times n} nonsingular matrix

    Show that rank(A)=rank(BA)=rank(AC)
    How you would prove this depends upon the precise definition of "rank" you are using. I am going to use this: If L is a linear transformation from vector space X to vector space Y, the rank(L) is the dimension of L(X).

    Let the rank of A be p. Then A maps R^n to a p-dimensional subspace of R^m. Since B is non-singular, it maps that p-dimensional subspace of [ R^m to a p-dimensional subspace of R^m. That is, BA maps R^n to a p-dimensional subspace of R^m. Therefore, rank(BA)= p.

    Since C is non-singular, it maps R^n to all of R^n. A then maps R^n to a p-dimensional subset of R^m. That is, AC maps R^n to a p-dimensional subset of R^m. Therefore, rank(AB)= p.
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