# Thread: Is this Linear Operator?

1. ## 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.

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

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

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

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