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Homework Help: Linear Transformations

  1. Apr 13, 2004 #1
    I'd like to check my proof. It seems easy enough, but I'd like to make sure that I'm not missing anything:

    If V is the space of all continuous functions on [0,1] and if
    Tf = integral of f(x) from 0 to 1 for f in V, show that T is a linear transformation From V into R1.

    Like I said the proof seems simple enough, but I just want to make sure I'm not missing anything that might be implied by "From V into R1."

    T(f + g) = integral from 0 to 1[f(x) + g(x)]dx
    = integral from 0 to 1 f(x)dx + integral from 0 to 1 g(x)dx
    = Tf + Tg

    T(kf) = integral from 0 to 1 kf(x)dx
    = k*integral from 0 to1 f(x)dx
    = kTf

    there for T is a linear transformation.

    I feel silly posting something this simple, but I'm just not absolutely sure that I'm not missing something.

    Thanks as usual for all the help.
  2. jcsd
  3. Apr 14, 2004 #2
    Yeah, that looks fine.

    [tex]f:V \rightarrow\ R[/tex] just refers to the fact that the definite integral over a function in V will always give you a constant real number.
  4. Apr 14, 2004 #3
    Thanks Stevo.
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