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Difference between linear and nonlinear transformation

  1. Nov 4, 2009 #1
    right now, my concept for their difference is that linear transformations are 1 to 1, where as nonlinear transformations are not. However, P^n to P^(-1) is a linear transformation, but it's not 1 to 1.

    the text book def of linear transformation is that it must be closed under addition and scale factorization.

    I'm confused.
     
  2. jcsd
  3. Nov 4, 2009 #2

    Office_Shredder

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    A linear transformation is just one that satisfies the conditions of linearity:

    T(x+y)=T(x)+T(y)
    T(ax)=aT(x)

    where x and y are vectors, a a scalar. Nothing requires linear transformations to be 1 to 1 (they usually aren't) or nonlinear transformations to not be 1-1
     
  4. Nov 5, 2009 #3

    HallsofIvy

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    Yes, you are. It makes no sense to talk about a transformation as being "closed under addition and scale factorization". Only sets are "closed" under given operations, not transformations. A "linear transformation" on a vector space is a function, F(v), such that F(au+ bv)= aF(u)+ bF(v). They are certainly NOT, in general, "one-to-one", as Office Shredder says. For example, any projection is linear but is not one-to-one.
     
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