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Consider the following operator: $L[f(x)]=\int_0^x f(s) ds$.

Is this operator linear?

I think in order to answer this question, I need to consider all possible functions, $f(x)$. The book says an operator is linear if you get a constant * the operator back: $L[f(x)]=cf(x)$.

I cannot integrate $f(s)ds$ without knowing what $f(s)$ is. So, suppose it is just $s$. Then the integral is $0.5s^2$ (evaluated from $0$ to $x$), or $0.5x^2$. This is not equal to $x$, so therefore the integral operator is not linear?

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    Are you *sure* the book says that? and not something more like, $L(cf)=cL(f)$?2012-11-21
  • 5
    ... and also $L(f+g) = L(f) + L(g)$.2012-11-21

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