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Math Help - Rank of Product Of Matrices

  1. #1
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    Rank of Product Of Matrices

    Question: Let A be an m * n matrix with rank m and B be an n * p matrix with rank n. Determine the rank of AB. Justify your answer.


    Attempt: I don't really know where to start off, but I have some things that might help me. I know that the rank of a matrix is equal to the number of linearly independent rows in it, and I also know that if A and B are two matrices, then rank(AB) <= rank(A) and also rank(AB) <= rank(B).
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  2. #2
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    I would do it like this:
    A is a mxn matrix of rank m, now it is obvius that m\leq n, why?
    So A represents a linear map f with rank m\leq n.
    B is a nxp matrix of rank n, it is equally obvius that p\geq n.
    So B represents a linear map g with of rank n.

    Now AB represents linear f\circ g, or f(g(\vec v))
    So g has a domain of p dimensions, and the codomain has n dimensions.
    f has domain of n dimensions, and the codomain has m dimensions.

    So AB represents a map f\circ g:V\to W, where V is p dimensional and W is m dimensional. So AB has rank m.

    This is what I would do, but there are probably nicer ways. Assuming ofcourse that this is correct.
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