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Cyclic decompostions

  1. Feb 27, 2012 #1
    Let T be a linear operator on the the finite dimensional space V, and let R be the range of T.

    (a) Prove that R has a complementary T-invariant subspace iff R is independent of the null space N of T.

    (b) If R and N are independent, prove that, N is the unique T-invariant subspace complementary to R.

    I supposed R has a complementary T - invariant subspace, say, W. Then , R should be T- admissible. I assumed to the contrary, that R intersection T is not equal to {0}. I took a point in the intersection but could not proceed further. Please suggest.
  2. jcsd
  3. Feb 27, 2012 #2
    Can you define "T-admisible"?
  4. Feb 27, 2012 #3
    Thanks for your reply.

    Given a linear operator T on a vector space V then a subspace W is T- admissible if

    i) W is invariant under T
    (ii) if f(T) β belongs to W, there exists a vector η in W such that f(T)β = f(T) η.
  5. Feb 27, 2012 #4
    What is f(T) in (ii)?
  6. Feb 28, 2012 #5


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    Hint: Fix a basis for the range of T and extend it to a basis for V. What can you say about these extra basis vectors?
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