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Math Help - Linear Transformations

  1. #1
    Senior Member tukeywilliams's Avatar
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    Linear Transformations

    I am trying to show that the R^2 --> R is NOT a linear transformation

    So I let (x,y) be a vector in R^2. Then the transformation is:

    T(x,y) = x

    T(c(x,y)) = T((cx,cy)) = cx = cT(x,y)

    Let (x1, y1) and (x2, y2) be vectors in R^2. Then T((x1,y1) + (x2,y2)) = T(x1+x2, y1+y2) = x1 + x2 = T(x1,y1) + T(x2,y2). But this should be false. What am I doing wrong?


    The same for R^2 --> R^2 (not a linear transformation)
    I let (x,y) be in R^2 and c (a scalar) be in R. Then:

    T(x,y) = (x,y)
    T(c(x,y)) = T(cx,cy) = cT(x,y). But this should be false according to the book. Should I even be choosing vectors at all?

    The last step is what I got, which should conclude that it is not a linear transformation. However, clearly T(u + v) = T(u) + T(v). Did I choose my vectors wrong?
    Last edited by tukeywilliams; April 11th 2007 at 01:39 AM.
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  2. #2
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    Quote Originally Posted by tukeywilliams View Post
    I am trying to show that the R^2 --> R is NOT a linear transformation
    How is the transformation defined?
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  3. #3
    Senior Member tukeywilliams's Avatar
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    yeah I think I got it now (didnt see how the transformation was defined).

    How would you see if this is a linear transformation:

    http://l.wordpress.com/latex.php?lat...&fg=000000&s=0


    What exactly is that saying?
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  4. #4
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    How is that a mapping from R^2 to R?
    The customary definition of C[a,b] is the set of all continuous functions on [a,b].
    What definition do you have to use?
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