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## Homework Statement

Consider the following operators acting in the linear space of functions Ψ(x) defined on

the interval (∞,∞)

(a) Shift Ta: TaΨ(x)=Ψ(x+a), a is a constant

(b) Reflection (inversion) I: IΨ(x)=Ψ(x)

(c) Scaling Mc: McΨ(x)= √c Ψ(cx), c is a constant

(d) Complex conjugation K: KΨ(x)=Ψ∗(x)

Are these operators linear? Find their adjoint operators. Find their inverse operators

## Homework Equations

Linear operator if:

i) kT(f) = T(kf)

ii) T(f+k) = T(f) + T(k)

## The Attempt at a Solution

I don't understand how to apply the linear operator conditions to these problems. Could someone explain to me a) or an example? I don't see how I can claim or prove (if right) TaΨ(x+b)=Ψ(x+a+b)