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Math Help - linear transformation

  1. #1
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    linear transformation

    How do I show that the transformation, T, defined by

    Tf(x) = {f(x)d(x) where { is the integral from 0 to 1

    is a linear transformation from C, the space of all real-valued continuous functions on [0, 1], into R?

    This idea is intuitive, but I'm having a tough time proving it. Thanks for your help.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by ktcyper03 View Post
    How do I show that the transformation, T, defined by

    Tf(x) = {f(x)d(x) where { is the integral from 0 to 1

    is a linear transformation from C, the space of all real-valued continuous functions on [0, 1], into R?

    This idea is intuitive, but I'm having a tough time proving it. Thanks for your help.
    Is d(x) fixed? Also, is this T:\mathcal{C}[0,1]\to\mathbb{R}:f(x)\mapsto\int_0^1f(x)d(x)dx? What's the problem? For example
    T(f(x)+g(x))=\int_0^1 d(x)(f(x)+g(x))dx=\int_0^1d(x)f(x)+\int_0^1d(x)g(x  ) =T(f(x))+T(g(x))
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  3. #3
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    I think you mean simply T(f)= \int_0^1 f(x) dx.

    Just use the definition of "linear transformation"- If a and b are real numbers and f and g are in C[0,1] then T(af+ bg)= \int_0^1 (af(x)+ bg(x)dx= a\int_0^1 f(x)dx+ b\int_0^1 g(x)dx= aT(f)+ bT(g).
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