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Linear Algebra - Dimension of Kernel

  1. Feb 14, 2008 #1
    1. The problem statement, all variables and given/known data
    Suppose that U and V are finite-dimensional vector spaces and that S is in L(V, W), T is in L(U, V). Prove that

    dim[Ker(ST)] <= dim[Ker(S)] + dim[Ker(T)]

    2. Relevant equations
    (*) dim[Ker(S)] = dim(U) - dim[Im(T)]
    (**) dim[Ker(T)] = dim(V) - dim[Im(S)]

    3. The attempt at a solution
    I know that ST is in L(U, W), so dim[Ker(ST)] = dim(U) - dim[Im(ST)]. So now I need to show:

    dim(U) - dim[Im(ST)] <= dim(U) + dim(V) - dim[Im(T)] - dim[Im(S)]

    But that just boils down to showing that dim[Im(ST)] is greater than dim[Im(T)] + dim[Im(S)]. It seems like I didn't really get anywhere. What am I missing? Thanks for your help.
  2. jcsd
  3. Feb 14, 2008 #2


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    Staff Emeritus
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    Are you and rjw5002 taking the same course? My response to his identical question is here:
  4. Feb 14, 2008 #3
    We must be taking the same course. It's the second course in Linear Algebra at Penn State. I read your post where you said the following:

    So I understand how you show that T is a subspace of Ker(ST), but what do you mean by "in order that STv=0, must be in the null space of T. That is, v is in T-1(null space of V)."? I'm not really sure what this is saying.
  5. Feb 15, 2008 #4


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    Staff Emeritus
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    Did I say that? Of course, that's a typo. If Tv is such that S(Tv)= 0, then Tv is in the null space of S. So that v itself is in T-1(null space of S).
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