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Matrix rank question

  1. Jan 23, 2010 #1
    Hey Guys, Another matrice question
    1. The problem statement, all variables and given/known data
    Prove: Rk(A+B)[tex]\leq[/tex] Rk(A) +Rk(B)

    3. The attempt at a solution

    Rk(A+B) = Dim[R(A) + R(B)]
    Where R(A) is the row space of A
    we know that Dim[R(A)+R(B)] = Dim[R(A)] + Dim[R(B)] - Dim[R(A)[tex]\cap[/tex]R(B)]
    Which means that Dim[R(A)+R(B)] [tex]\leq[/tex] Dim[R(A)] + Dim[R(B)] iff Rk(A+B)[tex]\leq[/tex] Rk(A) +Rk(B)

    I heard a rumor that this can also be done with linear transformations, can anyone elighten me on that path?

    Is this correct?
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Jan 24, 2010 #2


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    Science Advisor

    If F is a linear transformation from U to V, then, given specific bases for U and V, there exist a matrix representing F so essentially we can interpret matrices as being linear transformations and vice versa. Any thing true of matrices is true of linear transformations.

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