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Linear transformations with functions

  1. May 2, 2010 #1
    For the linear transformation,

    [tex]T: R^2\rightarrow R^2, T(x,y) = (x^2,y)[/tex]

    find the preimage of..
    [tex]f(x)= 2x+1[/tex]

    I have no trouble with these types of problems when it comes to vectors that aren't functions. Any help would be appreciated! Thanks!

    ~Matt
     
  2. jcsd
  3. May 3, 2010 #2

    radou

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    This is no linear transformation, since T(α v) = T(α(x, y)) = T(αx, αy) = (α^2x^2, αy), which does not equal αT(v), for some v from R^2 and some α from R.
     
  4. May 3, 2010 #3

    HallsofIvy

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    Since T is mapping pairs of numbers into pairs of numbers, I assume that by "f(x)= 2x+1" you mean the set of pairs (x, 2x+1)[/math]

    So you are looking for (x, y) such that [itex]T(x, y)= (x^2, y)= (a, 2a+1)[/itex] for some number a. Okay, since [itex]x^2= a[/itex], x can be either [itex]\sqrt{a}[/itex] or [itex]-\sqrt{a}[/itex]. And, of course, y= 2a+ 1. That is, the preimage is the set in [itex]R^2[/itex] [itex]\{(x,y)| y= 2\sqrt{x}+ 1 or y= -2\sqrt{x}+ 1\}[/itex].
     
    Last edited: May 4, 2010
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