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Math Help - Is this Linear Operator?

  1. #1
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    Is this Linear Operator?

    I know an Operator A is linear if A(f+g)=Af+Ag and A(cf)=cAf

    Let L=linear space={f(x):x in [0,1],f'(x) exists}

    Define as as follow: Af(x)=f(x)-f'(x). Is this linear operator? explain your answwer.

    I thought operator can only be on the form d/dy

    thanks for any help.
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  2. #2
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    You can combine the two conditions and say that transformation is linear if  A\big(\alpha f(x) + \beta g(x)\big) = \alpha A\big(f(x)\big) + \beta A\big(g(x)\big)

     A\big(\alpha f(x) + \beta g(x)\big) = \alpha f(x) + \beta g(x) -  (\alpha f(x) + \beta g(x))'

     = \alpha f(x) + \beta g(x) - \alpha f'(x) - \beta g'(x) = \alpha \big(f(x)-f'(x)\big) + \beta\big(g(x) - g'(x)\big)

     = \alpha  A\big(f(x)\big) + \beta A\big(g(x)\big)
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