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The statement "Can you apply a linear operation to the input and output of a linear operator?", can be written:

if $g = L(f)$ then does $M(g) = L(M(f))$,
where $L, M$ are linear (LTI) operators, and $f, g$ are functions?

Here is an attempt at a proof that this statement is true:

Since all linear LTI operators on functions can be represented as a convolution, *, of the input function with the operator's impulse response, let $Li$ and $Mi$ be the impulse responses of the LTI operators $L$ and $M$.

So $g = Li * f$

and since convolution is associative and commutative

$M(g) = Mi * [Li * f] = Mi * Li * f = Li * f * Mi = Li * [Mi * f] = L(M(f))$

which proves a linear operation can be applied to both the input and output of a LTI linear operator!

My question is: Is this proof good? A book (with page numbers) or a published paper will be appreciated.

Notes:

This attempts to provide a proof for Can the same linear operation be applied to both the input and output of a linear operator?

It builds on the answer to Can all linear operators on functions be represented as a convolution of the input function with the operator's impulse response? ,

specifically the answer https://math.stackexchange.com/a/2246879/147776 .

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