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Tricky transformations

  1. Apr 5, 2005 #1
    We know that a transformation from V to W is linear if the following hold:
    1.) For every x, y in V, T(x+y) = T(x) + T(y)
    2.) For every x in V and for every a in R (real numbers), T(ax) = aT(x)

    I need two nonlinear transformations from R to R. One must satisfy #1 above and violate #2. The other must violate #1 and satisfy #2.
     
  2. jcsd
  3. Apr 5, 2005 #2

    Hurkyl

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    Sounds like homework, so I'm moving it there. What have you done so far on this problem?
     
  4. Apr 5, 2005 #3

    dextercioby

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    The second one is really simple.Just take a nonlinear operator

    [tex] T(x)=x^{2} [/tex]

    I'll let a mathematician deal with the difficult issue.

    Daniel.


    EDIT:The above is wrong.I'll let a mathematician deal with the whole problem.
     
    Last edited: Apr 5, 2005
  5. Apr 5, 2005 #4

    Hurkyl

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    So are you claiming that T(ax) = aT(x) for your function?
     
  6. Apr 5, 2005 #5

    dextercioby

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    Ooops,sorry,Hurkyl,didn't see it. :redface: That nasty quadratic breaks both if them...

    Daniel.
     
  7. Apr 5, 2005 #6
    I think T(u) = u + k works for #1 but not #2.

    I don't know about the other one.
     
  8. Apr 5, 2005 #7
    No, it doesn't. With that,

    [tex]T(u+v) = u+v+k \neq u+v+2k = T(u) + T(v).[/tex]
     
  9. Apr 5, 2005 #8

    Hurkyl

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    Here's a hint...

    Suppose T(x) satisfies #1, and that you know T(x). Then, you also know T(2x) and T(3x), right? What about T(x/2)? T(47x/163)?
     
  10. Apr 5, 2005 #9

    Hurkyl

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    PS, this is a standard method of attack, and it's a good way to learn what things "really" mean.

    The whole point is to learn precisely what property #1 tells you, so you can find out what you can "break" so that property #2 fails. (and vice versa)
     
  11. Apr 6, 2005 #10
    Data,

    Touche! What was I thinking?
     
  12. Apr 6, 2005 #11

    Hurkyl

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    It's a mistake we all make at least once -- the trick is to catch it before you tell anybody. :smile:
     
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