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Transformations - Kernel

  1. Jun 16, 2007 #1


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    1. The problem statement, all variables and given/known data
    Given transformations T_1, T_2:V->F where V is a vector space with the dimension n over the field F, T_1 , T_2 =/= 0. If N_1 = KerT_1 , N_2 = KerT_2 and N_1 =/= N_2 find dim(N_1 intersection N_2)

    2. Relevant equations

    dim(A+B) = dimA + dimB - dim(A intersection B)

    3. The attempt at a solution
    First of all, dimImT_1 = dimImT_2 = 1 so N_1 = N_2 = n-1.
    Also, N_1 + N_2 in V so
    dim(N_1 + N_2) = n-1+n-1-dim(N_1 intersection N_2)
    <= n and so we get that dim(N_1 intersection N_2) >= n-2.
    But since N_1 intersection N_2 in N_1 we get
    n-1 >= dim(N_1 intersection N_2) >= n-2. But since N_1 =/= N_2 obviously
    dim(N_1 intersection N_2) =/= n-1 and so dim(N_1 intersection N_2) =n-2.
    Is that right? Did I leave out any important step?
  2. jcsd
  3. Jun 16, 2007 #2


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    Why is that true? What if T_1 were the identity transformation? How do you conclude that the dimension of the image of an arbitrary non-trivial linear transformation is 1 no matter what n is?

    Think about this special case: V= R3, T_1(x,y,z)= (x,y,z) and T_2= (x, y, 0). What are N_1 and N_2?
  4. Jun 16, 2007 #3


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    Sorry, I meant that both T_1 and T_2 are from V to F:
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