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Math Help - Matrices as Linear Maps

  1. #1
    Senior Member AllanCuz's Avatar
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    Matrices as Linear Maps

    Hey Team,

    Let A be an M by N matrix, B be an N by P matrix

    a) Show that AB is the sum of n matrices each of rank at most 1
    b) If the rank of A is n, what is the rank of AB?
    I'm not sure where to start on part a, but part b I think I have a decent start.

     Rank(AB) = dim(R[AB])

    We know that A maps from Fn to Fm, and B maps from Fp to Fn. So we define the left multiplication of these guys as such,

    L_A: F^N \to F^M

    L_B: F^P \to F^N

    So to find the range of AB we can let Z be some arbitrary vector in F^P and see for what values is it good for

    dim(R[AB])=dim(L_A L_B (Z))

    We can note that  L_B(Z) is a subset of F^N so the following inequality results,

    dim(L_A L_B (Z)) \le dim(L_A (F^N)) = dim(A(F^N)) = Rank(A) = N

    We can further note that if L_B is a surjective map then its range is actually equal to F^N turning the inequality into equality.

    Was the above process correct for b, and if so, are there any hints for a?

    Thanks!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Matrices as Linear Maps

    Quote Originally Posted by AllanCuz View Post
    Hey Team,



    I'm not sure where to start on part a, but part b I think I have a decent start.

     Rank(AB) = dim(R[AB])

    We know that A maps from Fn to Fm, and B maps from Fp to Fn. So we define the left multiplication of these guys as such,

    L_A: F^N \to F^M

    L_B: F^P \to F^N

    So to find the range of AB we can let Z be some arbitrary vector in F^P and see for what values is it good for

    dim(R[AB])=dim(L_A L_B (Z))

    We can note that  L_B(Z) is a subset of F^N so the following inequality results,

    dim(L_A L_B (Z)) \le dim(L_A (F^N)) = dim(A(F^N)) = Rank(A) = N

    We can further note that if L_B is a surjective map then its range is actually equal to F^N turning the inequality into equality.

    Was the above process correct for b, and if so, are there any hints for a?

    Thanks!
    That looks right or me. For the second part, consider writing the matrix as a sum of column matrices.
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  3. #3
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    Re: Matrices as Linear Maps

    strictly speaking, such a sum does not exist, perhaps you meant a sum of matrices that are all 0's except for one column?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Re: Matrices as Linear Maps

    Quote Originally Posted by Deveno View Post
    strictly speaking, such a sum does not exist, perhaps you meant a sum of matrices that are all 0's except for one column?
    Obviously, I was letting the OP fill things in for themselves, you know?
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  5. #5
    MHF Contributor

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    Re: Matrices as Linear Maps

    well you know me, i'm sorta slow-witted....
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