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Math Help - Rank proof

  1. #1
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    Rank proof

    Would someone please show me how to show that:

    rank(A) + rank(B) \leq rank(C)

    Thank you.
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Re: Rank proof

    Quote Originally Posted by itpro View Post
    Would someone please show me how to show that:

    rank(A) + rank(B) \leq rank(C)

    Thank you.
    Define A, B and C.
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  3. #3
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    Re: Rank proof

    Quote Originally Posted by alexmahone View Post
    Define A, B and C.
    They are nxn matrices.
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  4. #4
    MHF Contributor alexmahone's Avatar
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    Re: Rank proof

    Quote Originally Posted by itpro View Post
    They are nxn matrices.
    They can't just be any n x n matrices, can they? What is the relationship among them?
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  5. #5
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    Re: Rank proof

    Quote Originally Posted by alexmahone View Post
    They can't just be any n x n matrices, can they? What is the relationship among them?
    Sorry, missed that: C = A + B

    My intuition for the proof is that rank(A) and rank(b) is at most n and since addition preserves the dimension they will be at most of the rank of a C
    as C rank is less or equal to n.
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  6. #6
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    Re: Rank proof

    what you want to show is that:

    rank(A) + rank(B) ≤ rank(A+B).

    but this is not true. suppose that A =

    [1 0]
    [0 0] and that B =

    [-1 0]
    [ 0 0].

    clearly, A and B both have rank 1, so rank(A) + rank(B) = 1 + 1 = 2.

    however, A+B is the 0-matrix, which has rank 0, and 2 is not less than or equal to 0.
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