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Linear algebra question

  1. Sep 24, 2013 #1
    1. The problem statement, all variables and given/known data
    R(M) and C(M) are the row and column spaces of M.
    Let A be an nxp matrix, and B be a bxq matrix.
    Show that C(AB) = C(A) when the orthogonal complement of R(A) + C(B) = R^p (i.e. the orthogonal complement of R(A) and C(B) span R^p).


    2. Relevant equations
    I know that the orthogonal complement of R(A) is the null spaceo f A.
    I also know that C(X'X) = C(X') but that doesn't help


    3. The attempt at a solution
    Not sure where to go... thanks.
     
  2. jcsd
  3. Sep 24, 2013 #2

    jbunniii

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    Are you sure you stated the question correctly? If ##A## is ##n \times p## and ##B## is ##b \times q##, then the product ##AB## isn't even defined unless ##p = b##.
     
  4. Sep 24, 2013 #3
    I"m sorry, i meant A is nxp and B is p xq.
     
  5. Sep 24, 2013 #4

    jbunniii

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    By "orthogonal complement of R(A) + C(B)" do you mean ##R(A)^{\perp} + C(B)##, or something else?
     
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