vittorecostao1

2022-06-19

Showing that the matrix transformation $T\left(f\right)=x\ast {f}^{\prime }\left(x\right)+{f}^{″}\left(x\right)$ is linear

Layla Love

By the definition of T, you have for every function h in your domain of T, that
$T\left(h\right)=x\cdot {h}^{\prime }\left(x\right)+{h}^{″}\left(x\right)$
If now $h=f+g$, we get
$T\left(f+g\right)=x\cdot \left(f+g{\right)}^{\prime }\left(x\right)+\left(f+g{\right)}^{″}\left(x\right),$which equals your calculated result, which moreover used the linearity of differentiation.

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