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Homework Help: Null spaces, transformations

  1. May 7, 2007 #1
    1. The problem statement, all variables and given/known data
    Let T:V  W be a linear transformation. Prove the following results.

    (a) N(T) = N(-T)
    (b) N(T^k) = N((-T)^k)
    (c) If V = W and t is an eigenvalue of T, then for any positive integer k
    N((T-tI)^k) = N((tI-T)^k) where I is the identity transformation

    3. The attempt at a solution
    (a) for every x in V:
    If T(x) = y, then –T(x) = -y
    So then, T(0) = 0 = -T(0)
    Is this right? On the right track?

    I’m not sure how to approach the rest of them?

    Thanks for your help!
     
  2. jcsd
  3. May 7, 2007 #2

    AKG

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    a) is right. b) is done essentially the same way. If Tkx = y, then (-T)kx = ___? It should be pretty easy. c) follows immediately from b).
     
  4. May 7, 2007 #3

    Hurkyl

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    Well, everything he said is right, but he hasn't proven what he set out to prove: that N(T) and N(-T) are equal sets.
     
  5. May 8, 2007 #4

    AKG

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    Sorry, my mistake. Hurkyl is correct, redyelloworange has not yet answered part a) fully.
     
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