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Linear transformation

  1. Jul 20, 2010 #1
    I'm hoping I can get some help with the following question:

    Does definite integration (from x = 0 to x = 1) of functions in Pn correspond to some linear transformation from Rn+1 to R?

    OK, well my original answer was yes, but the textbook says "no, except for P0" which I do not understand.

    So I have p(x) = anxn + an-1xn-1 + ... + a1x + a0

    If I integrate from 0 to 1, I get: P(1) = an/(n+1) + an-1/n + ... + a1/2 + a0


    So I have T(an, an-1, ..., a1, a0) = (an/(n+1) + an-1/n + ... + a1/2 + a0)

    and T: Rn+1 -> R

    And if I want to show that it is linear, then I show that the transformation has these properties:
    T(kx) = kT(x) and
    T(x+y) = T(x) + T(y)
    and I think both cases are obvious.

    And T is multiplication by [1/(n+1) | 1/n . . . 1/2 | 1]

    What am I missing?
    Thanks in advance for any help.
  2. jcsd
  3. Jul 20, 2010 #2


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    Homework Helper

    Linear transformations usually mean between vector spaces of the same dimension and therefore can be loosely thought of as matrices. what you have is a linear functional.

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