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Range space of linear mappings

  1. May 9, 2009 #1
    1. The problem statement, all variables and given/known data
    Let L : Rn --> Rm and M : Rm --> Rp be linear mappings.
    a)Prove that rank( M o L) <= rank(L).
    b)Give an example such that the rank(M o L) < rank(M) and rank(L)

    2. Relevant equations
    None


    3. The attempt at a solution
    a)I see that (M o L) takes all vectors in Rn and maps them to vectors in Rm then maps these vectors to vectors in Rp. (L) also takes all vectors in Rn and maps them to Rm. From this, i get the impression that rank(M o L) = rank (L) because the quantity of vectors should not change when (M o L) maps vectors in Rm to Rp.

    b)Is there a method to get such a matrix or do I have to use trial and error?
     
  2. jcsd
  3. May 9, 2009 #2
    Hints:

    For part a:
    Note that the range of M is a subspace of Rp with dimension Rank(M).
    Likewise, the range of L is a subspace of Rm with dimension Rank(L).
    For the composition ML, notice that after L is applied, the range of L is not necessarily all of Rm. Moreover, when you next apply M, it is only acting on that subspace, range of L.

    For part b, try matrices from R^2 to R^2. Make both of them with rank 1, yet the composition has rank 0.
     
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