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Establish criteria for f in the Range(L)

  1. Feb 26, 2012 #1
    So..here's the problem:

    Let Z={x: {0,1,..,5} → ℝ^n} (column vector with real entries) and define T:Z→ℝ^n by Tz = z(0) - z(5). Let X = Ker(T) and let Y={y: {0,1,..,4}→ℝ^n}. Define L:X→Y by
    (Lx)(t) = x(t+1) - Ax(t) where A is an invertible matrix.

    Establish criteria for f in Range(L).

    I know for f to be in Range(L) there exists x in X such that L(x) = f
    so (Lx)(t) = f(t) for all t,
    then x(t+1) - Ax(t) = f(t) for all t,
    or x(t+1) = Ax(t) + f(t).

    Then from what I was told I want to look at what x(t) looks like for x = 0,1,2,...5 and determine what needs to be true of f. But that's all the further I can get, any advice?
     
  2. jcsd
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